Log In Start studying!

Select your language

Suggested languages for you:
Deutsch (DE)
Deutsch (UK)
Deutsch (US)

Americas

Europe

Answers without the blur. Sign up and see all textbooks for free! → Illustration

Q49E

Expert-verified
Essential Calculus: Early Transcendentals
Found in: Page 380
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later
Illustration

Short Answer

(a) To determine the Cavalieri's Principle.

(b) To determine the volume of the oblique cylinder using Cavalieri's principle.

(a) The Cavalieri's Principle is proved.

(b) The volume of the oblique cylinder by the use of Cavalieri's principle is \(\pi {r^2}h\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Given

(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids \({S_1}\) and \({S_2}\), then the volume of \({S_1}\) and \({S_2}\) are equal.

(b) The oblique cylinder of radius \(r\) and height \(h\).

The Concept ofmoment and forces

The Summation of moment about any point is Equal to zero.

The Summation of forces along any direction is equal to zero.

Evaluate the Cavalieri's Principle (a)

The expression to find the volume of the parallel planes as shown below:

\(V = \int_0^h A (z)dz\) …………….. (1)

Refer to Equation (1).

Consider that the volume of solid \({S_1}\)and \({S_2}\) are equal.

\(\begin{aligned}{}V\left( {{S_1}} \right) = V\left( {{S_2}} \right)\\ = \int_0^h A (z)dz\end{aligned}\)

Hence, the Cavalieri's Principle is proved.

Evaluate the volume of the oblique cylinder(b)

The area of the oblique cylinder as shown below.

\(A(z) = \pi {r^2}\)

Find the volume of the oblique cylinder by the use of Cavalieri's principle as shown below.

Substitute \(\pi {r^2}\) for \(A(z)\) in Equation (1).

\(\begin{aligned}{}V = \int_0^k \pi {r^2}dz\\ = \pi {r^2}(z)_0^h\\ = \pi {r^2}(h - 0)\\ = \pi {r^2}h\end{aligned}\)

Therefore, the volume of the oblique cylinder by the use of Cavalieri's principle is \(\underline {\pi {r^2}h} \).

Most popular questions for Math Textbooks

Find the area of the shaped region.

Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use Simpson's Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

y=sin2 x about \(y = - 1\)

y=0,0≤x≤π

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

\( 10. 4x + {y^2} = 12,\quad x = y\)

(a) To determine the integral for the volume of the solid obtained by rotating the region bounded by the given curve.

(b)To determine the value of integral function.
Icon

Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

Decision Maths

Read explanation

Calculus

Read explanation

Probability and Statistics

Read explanation

Statistics

Read explanation

Mechanics Maths

Read explanation

Geometry

Read explanation

Pure Maths

Read explanation

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.