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Determining Volumes by Slicing

Determining Volumes by Slicing

Ever tried home-made potato chips? You can try doing some! All you need is a knife and a deep frier (and of course, a potato, salt, and oil). What characterizes potato chips from french fries is the shape of the cut. For a potato chip you need to do thin slices, so they can fry properly.

After slicing the potato you can begin to think backwards. Can you put all the slices back together to make the potato again? Maybe this is a little hard to do because, once sliced, you cannot reconstitute back the potato. However, the volume of the slices is equal to the volume of the potato before you sliced it.

The idea of slicing is very useful to find volumes because if the slices are very thin, you can take advantage of formulas for finding areas instead, and then reconstitute back the solid by means of integration.

What is the Meaning of Determining Volumes by Slicing?

You have probably found before formulas for finding the volumes of some geometric bodies, for example if you are given a cone, you can find its volume using the formula

\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h,\]

or maybe you are given a cylinder, whose volume can be found with

\[ V_{\text{cylinder}} = \pi r^2 h.\]

There is also the volume of a pyramid, which regardless of the shape of its base, you can find its volume with

\[ V_{\text{pyramid}} = \frac{1}{3}A h.\]

But where do all these formulas come from? Why are there \( \frac{1}{3} \) involved?

The answer to this question becomes easier if you slice any of these solids, obtaining a cross-section of the body.

A cross-section of a body is a view that shows how the body would look if you were to make a cut across it.

Consider the case of a cone. Here you can see a cross-section obtained by slicing the cone halfway through its height. The slice is done so it is parallel to the base of the cone.

Determining Volumes by Slicing cone cross-section parallel to its base StudySmarterFigure 1. The cross-section parallel to the base of a cone is a circle

This is particularly useful because the formula for finding the area of a circle is

\[ A_{\text{circle}} = \pi r^2, \]

so you can add together all the cross-sections of the cone by means of integration to obtain the volume of a cone.

Finding Volumes by Slicing

To find volumes by slicing, consider a simpler example: the volume of a rectangular prism.

Determining Volumes by Slicing rectangular prism StudySmarter
Figure 2. A rectangular prism

Next, slice the prism to obtain a cross-section that is parallel to any of its faces, for example a cross-section parallel to its base.

Determining Volumes by Slicing cross-section of a rectangular prism StudySmarterFigure 3. The cross-section parallel to the base of a rectangular prism is a rectangle

You can find the area of a rectangle by using the formula

\[ A_{\text{rectangle}} = \ell w.\]

To find the volume of the rectangular prism, imagine you stack all the rectangular slices of the prism. Since all slices are equal, you just multiply the area of the rectangular prism (let's just call it \( A \) for simplicity) by the height of the prism, that is

\[ \begin{align} V_{\text{prism}} &= A h \\ &= \ell w h. \end{align} \]

You can obtain the same formula if stacking cross-sections that are parallel to any other side of the prism!

You can imagine this step as if dragging the area of a cross-section of the prism from one side to another

Determining Volumes by Slicing rectangular prism volume StudySmarterFigure 4. Volume of the rectangular prism obtained by dragging the area of a cross-section from one side to another

A similar idea applies to other geometric bodies, like cylinders.

Determining Volumes by Slicing cylinder volume StudySmarterFigure 5. Volume of a cylinder obtained by dragging the area of a cross-section from a base to another

The volume of a cylinder can be found by multiplying the area of the base by its height, so

\[ \begin{align} V_{\text{cylinder}} &= A h \\ &= \pi r^2 h. \end{align}\]

But what happens if the slices are not equal? You need to integrate!

Determining Formulas for Volumes by Slicing

So far you have seen that if all the areas of a cross-section of a solid are equal, you can just multiply by the length of the corresponding transversal side to obtain the volume of a solid, like in the case of the rectangular prism or with the cylinder.

Now consider the case of a cone. Suppose that the base of this cone is equal to \(R,\) and its height is \( h.\) If you take cross-sections parallel to its base, you will find out that these do not have the same area.

Determining Volumes by Slicing cross-sections of a cone have different areas StudySmarterFigure 6. The cross-sections of a cone do not have the same area

For illustrative purposes, place the cone along the \(x-\)axis with its tip in the origin. This way you can find the radius of each cross-section as a function of \(x.\)

Determining Volumes by Slicing radius of a cone viewed from a side StudySmarterFigure 7. The radius of each cross-section is a function of \(x.\)

In fact, the cross-sections are all circles with different radii, which can be found by using similar triangles, that is

\[ \frac{R}{h} =\frac{r}{x},\]

so

\[ r = \frac{R}{h}x.\]

Determining Volumes by Slicing similar triangles used to find the radius of a cross-section of a cone StudySmarterFigure 8. You can find the radius of a cross-section with the help of similar triangles

The function

\[ r(x) = \frac{R}{h}x\]

gives you the radius of each cross-section in terms of its position relative to the tip of the cone, so the area of a cross-section is

\[ \begin{align} A(x) &= \pi \left( r(x) \right) ^2 \\ &= \pi \left( \frac{R}{h}x \right)^2 \\ &= \frac{\pi R^2}{h^2}x^2. \end{align} \]

Now that you know a function for the area of each cross-section, you can add them together by means of integration. The integration limits are \(a=0\) since you are starting from the tip of the cone, and \( b=h\) because the cross-sections align all the way up to the base, so

\[ \begin{align} V &= \int_0^h A(x) \,\mathrm{d}x \\ &= \int_0^h \frac{\pi R^2}{h^2}x^2\,\mathrm{d}x \\ &= \frac{\pi R^2}{h^2}\int_0^h x^2\,\mathrm{d}x. \end{align}\]

You can solve the resulting definite integral using the Power Rule and the Fundamental Theorem of Calculus, that is

\[ \begin{align} \int_0^h x^2 \, \mathrm{d}x &= \left( \frac{1}{3}(h)^3 \right) - \left( \frac{1}{3}(0)^3\right) \\ &= \frac{1}{3}h^3. \end{align} \]

By plugging back the above definite integral into the formula for the volume, you obtain

\[ \begin{align} V &= \left( \frac{\pi R^2}{h^2} \right) \left(\frac{1}{3}h^3 \right) \\ &= \frac{\pi R^2h^3}{3h^2} \\ &= \frac{1}{3}\pi R^2 h,\end{align}\]

which is the formula for the volume of a cone,

\[V_{\text{cone}} = \frac{1}{3}\pi R^2h.\]

So that's where the \( \frac{1}{3} \) comes from! You can find formulas for more solids by following steps similar to above.

Determining Volumes by Slicing Solids of Revolution

Another way of finding volumes of solids is by rotation of two-dimensional figures around a revolution axis. The solids found this way are known as Solids of Revolution, and their volume are found using different methods depending on the solid.

Determining Volumes by Slicing solid of revolution of a parabola StudySmarterFigure 9. A solid of revolution obtained by rotation of a parabola along the \(x-\)axis

Check out our articles about the Disk Method and the Washer Method for more information on this subject!

Examples of Volumes Determined by Slicing

There are more formulas that can be determined by slicing!

Find the formula for the volume of a square pyramid with side \( \ell \) and height \( h.\)

Answer:

The base of a square pyramid, as the name suggests, is square, so its area can be found using the formula for the area of a square, that is

\[ A_{\text{square}}=\ell ^2.\]

The cross-sections parallel to the base of the pyramid are smaller squares. You can use similar triangles to find that the length \( s \) of the sides of these squares are given by

\[ s(x) = \frac{\ell}{h}x. \]

Knowing this, the area of each cross-section becomes

\[ \begin{align} A(x) &= \left( s(x) \right)^2 \\ &= \left( \frac{\ell}{h}x\right)^2 \\ &= \frac{\ell^2}{h^2}x^2. \end{align} \]

By integrating this area along the \(x-\)axis from \( 0 \) to \( h, \) you can find the volume of the pyramid, so

\[ \begin{align} V &= \int_0^h A(x) \, \mathrm{d}x \\ &= \int_0^h \frac{\ell^2}{h^2} x^2 \, \mathrm{d}x \\ &= \frac{\ell^2}{h^2} \int_0^h x^2 \, \mathrm{d}x. \end{align}\]

The resulting definite integral is the same as the previous example of the cone, you can find it with the help of the Power Rule and the Fundamental Theorem of Calculus, giving you

\[ \int_0^h x^2 \, \mathrm{d}x = \frac{1}{3}h^3.\]

By plugging this integral back into the volume formula, you obtain

\[ \begin{align} V &= \left( \frac{\ell^2}{h^2} \right) \left( \frac{1}{3}h^3 \right) \\&= \frac{\ell^2h^3}{3h^2} \\ &= \frac{1}{3}\ell^2h. \end{align} \]

Note that the area of the base is \( A=\ell^2,\) so you can substitute this into the above formula and obtain

\[ V=\frac{1}{3}Ah,\]

which might look more familiar.

Ever wondered why the formula for the volume of a sphere has \( \frac{4}{3} \) in it?

Find the formula for the volume of a sphere with radius \( R.\)

Answer:

Begin by imagining you place the sphere in the origin of the coordinate system. The cross-sections of a sphere are all circles, so you can find the area of a cross-section of the sphere using

\[ A(x) = \pi \left(r(x)\right)^2.\]

You need to find the radius of the sphere depending on its position relative to the origin. In order to do so, focus on the cross-section of the sphere.

Determining Volumes by Slicing radius of a sphere cross section in terms of x  StudySmarterFig. 10. The radius of a cross-section of the sphere with radius R can be expressed in terms of \(x\).

The radius of a cross-section satisfies the equation

\[x^2+r^2=R^2,\]

from where you can obtain

\[r^2=R^2-x^2,\]

so

\[\left( r(x) \right) ^2 = R^2-x^2.\]

This means that the area of a cross-section is given by

\[ A(x) = \pi (R^2-x^2),\]

so you can integrate to find the volume of the sphere, that is

\[ V(x) = \int_{-R}^R \pi \left(R^2-x^2\right) \, \mathrm{d}x\]

The integral can be done with the Power Rule. Do not forget that in this case, R is a constant!

\[ \begin{align} V(x) &= \int_{-R}^R \pi \left( R^2-x^2\right) \, \mathrm{d}x \\ &= \pi \left[ \left( R^2(R)-\frac{1}{3}(R)^3 \right) - \left( R^2(-R) -\frac{1}{3}(-R)^3 \right) \right] \\ &= \pi \left[ \left(R^3-\frac{1}{3}R^3 \right)-\left( -R^3+\frac{1}{3}R^3 \right)\right]\\ &=\pi \left[ \frac{2}{3}R^3+\frac{2}{3}R^3\right] \\ &= \frac{4}{3}\pi R^3. \end{align}\]

And that is how you obtain the formula for the volume of a sphere!

Determining Volumes by Slicing - Key takeaways

  • A cross-section of a body is a view that show how the body would look if you were to make a cut across it.
  • The idea behind determining volumes by slicing is to find the areas of the slices of a body, and integrate them to obtain their volume.
    • In order to do so, you obtain slices, or cross-sections, aligned along the axis of integration.
  • The formulas for many geometrical bodies can be found by slicing, like the volumes of pyramids, cones, and spheres.

Frequently Asked Questions about Determining Volumes by Slicing

Slicing a volume, or slicing a body, refers to the fact that you can divide a body into simpler shapes, like disks or rectangles. This allows you to obtain formulas for the volume of the bodies by means of integration.

  1. Find a cross-section of the body that has a regular shape, so you can find its area.
  2. Obtain an expression for the area of the cross-section.
  3. Integrate along an axis perpendicular to the planes that contain the cross-sections.

Suppose you have a three-dimensional body. If you slice it you will obtain a cross-section. By identifying the shape of the cross-section you can use an appropriate formula to find its area.

To obtain the volume of a body by slicing you need to integrate the area of a cross-section along an axis that is perpendicular to the planes containing the cross-sections.

An example is the volume of a sphere. If you integrate the area of the circles that make up its cross-sections, you obtain the volume of the sphere.

Final Determining Volumes by Slicing Quiz

Question

What is a solid of revolution?

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Answer

solid of revolution is a solid figure obtained from rotating a curve around a straight line.

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Question

What is a surface of revolution?

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Answer

The surface of revolution is the surface which bounds the solid of revolution.

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Question

Given a function \(f(x),\) how can you obtain a solid of revolution?

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Answer

To obtain a solid of revolution starting with a function \( f(x) \) over an interval \( [a,b],\) rotate the curve about a given axis (vertical or horizontal), producing a three-dimensional region.

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Question

How can you obtain the volume of a solid of revolution using the disk method?

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Answer

You need to slice the solid of revolution in disks, and then add the volume of all the disks.

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Question

Rotating a function around the \(z-\)axis produces disks (or washers) that are parallel to ____.

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Answer

the \(xy-\)plane.

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Question

What are the two major methods for calculating the volume of a solid of revolution?

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Answer

Disk and washer methods

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Question

What is the difference between the disk and washer methods?

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Answer

The disk method sums the area of infinitely many circular disks while the washer method sums the area of infinitely many circular washers (a disk with a hole in the middle).

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Question

How do you know when to use the disk method?

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Answer

The disk method should be used when the axis which the curve revolves around is also a boundary for the solid of revolution.

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Question

How do you know when to use the washer method?

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Answer

When the axis which the curve revolves around is not a boundary for the solid of revolution, the washer method is used.

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Question

Rotating a function around the \(y-\)axis produces disks (or washers) that are parallel to ____.

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Answer

the \(xz-\)plane.

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Question

Rotating a function around the \(x-\)axis produces disks (or washers) that are parallel to ____.

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Answer

the \(yz-\)plane.

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Question

What is the disk method used for?

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Answer

The disk method is used for finding the volume of a solid of revolution.

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Question

The disks used in the disk method are contained in planes that are ____ to the axis of rotation.

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Answer

perpendicular.

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Question

What is the disk method within the context of calculus?

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Answer

The disk method is a method for calculating the volume of a solid of revolution that is used when integrating along an axis that is parallel to the axis of integration.

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Question

The disk method divides the region into very thin ____, also known as disks.

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Answer

cylinders.

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Question

Suppose a solid of revolution is obtained by the rotation of a region around the \(y-\)axis. Which formula should you use?

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Answer

\[ V=\int_a^b \left[f(y)\right]^2\,\mathrm{d}y\].

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Question

Suppose a solid of revolution is obtained by the rotation of a region around the \(x-\)axis. Which formula should you use?

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Answer

\[ V=\int_a^b \left[f(x)\right]^2\,\mathrm{d}x\].

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Question

The difference between the shell method and the disk method is ____.

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Answer

the shape in which the solid is divided.

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Question

What is the washer method used for?

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Answer

The washer method is a method for calculating the volume of a solid of revolution when the axis of revolution is not a bound of the revolution region.

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Question

When should one use the washer method?

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Answer

The washer method should be used when the revolving region is not bounded by the axis of revolution.

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Question

Where did the washer method get its name?

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Answer

It is called the "washer method" because the cross-sections look like washers.

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Question

The washer method is a modification of ____.

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Answer

the disk method.

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Question

Which of the following is the formula for the washer method?

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Answer

\[ V=\int_a^b \pi (R^2-r^2)\,\mathrm{d}x\].


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Question

Which of the following formulas gives the area of a cross-section of a washer?

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Answer

\[ A = \pi (R^2-r^2) \].

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Question

The washer method always produces hollow solids of revolution.

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Answer

true.

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Question

The centers of every washers are aligned along ____.

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Answer

the axis of revolution.

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Question

If the region is revolved around a horizontal line, you should integrate with respect to ____.

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Answer

x.

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Question

If the region is revolved around a vertical line, you should integrate with respect to ____.

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Answer

y.

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Question

The radius \( R \) of the outer circle is the function that is ____ the revolution axis.

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Answer

farther from.

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Question

The radius \( r \) of the outer circle is the function that is ____ the revolution axis.

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Answer

closer to.

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Question

The washers can also be seen as ____.

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Answer

donuts.

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Question

Can you use the disk method if a region is bounded between two curves?

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Answer

No, the washer method is used in that scenario.

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Question

What is a cross-section of a body?

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Answer

A cross-section of a body is a view that shows how the body would look if you were to make a cut across it.

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Question

Suppose you slice a cylinder in a way such that the slice is parallel to its base. What shape is the resulting cross-section?

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Answer

circular.

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Question

Suppose you slice a cylinder in a way such that the slice is perpendicular to its base. What shape is the resulting cross-section?

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Answer

rectangular.

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Question

Suppose you slice a triangular pyramid in a way such that the slice is parallel to any of its faces. What shape is the resulting cross-section?

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Answer

triangular.

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Question

Suppose you want to slice a rectangular pyramid in a way such that the cross-section is rectangular. In order to do so, the slice has to be ____ to the base.

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Answer

parallel.

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Question

The volume of a ____ can be found by multiplying the area of its base by its height.

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Answer

cylinder.

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Question

The volume of a ___ can be found by multiplying the area of its base by its height.

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Answer

rectangular prism.

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Question

In order to find the volume of a solid, you need to ____ the area of a cross-section along an axis perpendicular to the planes that contain the cross-sections.

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Answer

integrate.

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Question

Is it possible to obtain a triangular cross-section by slicing a cone?

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Answer

Yes.

Show question

Question

The cross-sections parallel to any face of a cube are all square.

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Answer

True.

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Question

Any cross-section of a cube is either a square or a rectangle.

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Answer

False.

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Question

The line used as a reference for the rotation of a curve in a solid of revolution is called the axis of ____.

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Answer

revolution.

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Question

True/False: The axis of revolution has to be either the x-axis or the y-axis.

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Answer

False.

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Question

True/False: The axis of revolution can be the curve \( y=x^2.\)

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Answer

False.

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Question

True/False: The axis of revolution can be the curve \( y=2x.\) 

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Answer

True.

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Question

True/False: A surface of revolution has no volume.

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Answer

True.

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Question

True/False: The axis of revolution has to pass through the origin.

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Answer

False.

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Question

A solid of revolution is also known as a(n) ____.

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Answer

volume of revolution.

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