Answered: y2 = x(4 – x)² 4 3 2 1 ++ x 1 2 3/45 6…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the graph of y² = x(4 − x)² , as shown in the figure. Find the volumes of the solids that are generated when the loop of this graph is revolved about (a) the x-axis, (b) the y-axis, and (c) the line x = 4.

y2 = x(4 – x)²
4
3
2
1
++ x
1 2 3/45 6 7
-1
-2 +
-3
-4+

Expert Solution

Step 1

Given: $y^{2} = x {(4 - x)}^{2}$

To find the volume of the solid revolved about

(i) x-axis

(ii) y-axis

(iii) The line x=4.

Step 2

We have the curve $y^{2} = x {(4 - x)}^{2}$ .

(a) The volume of the solid generated by revolution about the x-axis is calculated by the following formula:

$V = π \int_{a}^{b} {[f (x)]}^{2} d x$

It is known as the disk method.

Thus, the volume of the solid is,

$V = π \int_{0}^{4} x {(4 - x)}^{2} d x \Rightarrow V = π \int_{0}^{4} x (16 - 8 x + x^{2}) d x \Rightarrow V = π \int_{0}^{4} (16 x - 8 x^{2} + x^{3}) d x \Rightarrow V = π {(\frac{16 x^{2}}{2} - \frac{8 x^{3}}{3} + \frac{x^{4}}{4})}_{0}^{4} \Rightarrow V = π {(8 x^{2} - \frac{8 x^{3}}{3} + \frac{x^{4}}{4})}_{0}^{4} \Rightarrow V = π (8 {(4)}^{2} - \frac{8 {(4)}^{3}}{3} + \frac{{(4)}^{4}}{4}) \Rightarrow V = \frac{64}{3} π$

(b)The volume of the solid generated by revolution about the y-axis is calculated by the following formula:

$V = 2 π \int_{a}^{b} x f (x) d x$

It is known as the shell method.

Thus, the volume is,

$V = 2 π \int_{0}^{4} x \sqrt{x} (4 - x) d x \Rightarrow V = 2 π \int_{0}^{4} x^{\frac{3}{2}} (4 - x) d x \Rightarrow V = 2 π {[\frac{4 x^{\frac{5}{2}}}{\frac{5}{2}} - \frac{x^{\frac{7}{2}}}{\frac{7}{2}}]}_{0}^{4} \Rightarrow V = 2 π {[\frac{8 x^{\frac{5}{2}}}{5} - \frac{2 x^{\frac{7}{2}}}{7}]}_{0}^{4} \Rightarrow V = 2 π {[\frac{8 {(4)}^{\frac{5}{2}}}{5} - \frac{2 {(4)}^{\frac{7}{2}}}{7}]}_{0}^{4} \Rightarrow V = \frac{1024}{35} π$

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