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Q1E

Expert-verifiedFound in: Page 384

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**(a)To determine the difficulty to use slicing to find the volume, \(V\) of solid \(S\).**

**(b)To sketch the typical approximating shell.**

**(c)To find the circumference, height and volume using the method of shell.**

The circumference, height and volume of shell as follows, \(C = 2\pi x\;,\;\;H = x{(x - 1)^2}\), and \(V = \frac{\pi }{{15}}\).

The function is \(y = x{(x - 1)^2}\)

The region lies between 0 to 1.

**The volume by the use of the method of shell is **\(V = \int_a^b 2 \pi x(f(x))dx\)

Show the equation as below:

\(y = x{(x - 1)^2}\) ……… (1)

Plot a graph for the equation \(y = x{(x - 1)^2}\)by the use of the calculation as follows:

Calculate \(y\) value by the use of Equation (1).

Substitute 0 for \(x\) in Equation (1).

\(\begin{aligned}{}dy = 0{(0 - 1)^2}\\ = 0\end{aligned}\)

Hence, the co-ordinate of (x, y) is \((0,0)\).

Calculate \(y\) value by the use ofEquation (1).

Substitute 1 for \(x\) in Equation (1).

\(\begin{aligned}{}dy = 1{(1 - 1)^2}\\ = 0\end{aligned}\)

Hence, the co-ordinate of (x, y) is \((1,0)\).

Draw the region as shown in Figure

Draw the shell

Refer Figure 2

The radius of shell is \(x\).

Calculate the circumference of shell:

\(C = 2\pi r\) ………. (2)

Substitute \(x\) for \(r\) in Equation (2).

\(C = 2\pi x\)

Calculate the height of shell:

\(H = y\) ..……. (3)

Substitute \(x{(x - 1)^2}\) for \(y\) in Equation (3).

\(H = x{(x - 1)^2}\)

Calculate the volume by the use ofthe method ofshell:

\(V = \int_a^b 2 \pi x(f(x))dx\) …….. (4)

Substitute0 for a, 1 for \(b\), and \(x{(x - 1)^2}\) for \((f(x))\) in Equation (4)\(V = \int_0^1 2 \pi x\left( {x{{(x - 1)}^2}} \right)dx\)

\(V = 2\pi \int_0^1 {{x^2}} \left( {{x^2} - 2x + 1} \right)dx\) …….. (5)

Integrate Equation (5).

\(\begin{aligned}{}V = 2\pi \left( {\frac{{{x^{4 + 1}}}}{{4 + 1}} - \frac{{2{x^{3 + 1}}}}{{3 + 1}} + \frac{{{x^{2 + 1}}}}{{2 + 1}}} \right)_0^1\\ = 2\pi \left( {\frac{{{x^5}}}{5} - \frac{{{x^4}}}{2} + \frac{{{x^3}}}{3}} \right)_0^1\\ = \frac{\pi }{{15}}\end{aligned}\)

Therefore, the circumference, height and volume of shell as follows, \(C = 2\pi x,\;\;\;H = x{(x - 1)^2}\), and \(V = \frac{\pi }{{15}}\).

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