Question

Volumes of the solids with disk or shell method?

Answer 1

The answer to the questions of volumes are given as follows

a) $v=128 \pi$

b) $v=\frac{128}{3} \pi$

 c)$v=\frac{1024}{5} \pi$

d)$V=\frac{13568}{15} \pi$

Generally, the questions are  mathematically solved below

$y=\sqrt{x}, y=4, x=0$

a) x-axis

if $y=4, x=16, x=0$

Using disk method

$} v=\pi \int_{0}^{16}(\sqrt{x})^{2} d x$

$v=\pi\left(\frac{x^{2}}{2}\right)_{0}^{16}$

$v=\frac{\pi}{2} \times 16 \times 16$

$v=128 \pi$

b)  line y=4

if x=0, y=0 ;

$y=\sqrt{x} \Rightarrow x=y^{2}$

Using shell method

$v = \int_{0}^{4} 2 \pi(4-y) \cdot y^{2} d y$

$v=2 \pi \int_{0}^{4}\left(4 y^{2}-y^{3}\right) d y$

$v=2 \pi\left[\frac{4 y^{3}}{3}-\frac{y^{4}}{4}\right]_{0}^{4}$

$v=\frac{2 \pi}{12}[1024-768]$

$v=\frac{512 \pi}{12}$

$v=\frac{128}{3} \pi$

c) y-axis

0 ≤ y ≤ 4

x=y^2

Using disk method

volume

$v=\pi \int_{0}^{4} y^{4} d y$

$v=\pi\left(\frac{y 5}{5}\right)_{0}^{4}$

$v=\frac{1024}{5} \pi$

d) line x=-1

y=√x, y=4, x=0

0 ≤ x ≤ 6

Using shell method

volume is

$V=\int_{0}^{16} 2 \pi(1+x) \sqrt{x} d x$

$V=2 \pi\int_{0}^{16}(x^{1 / 2}+x^{3 / 2}\right) )d x\right.$

$V=2 \pi\left[\frac{x^{3 / 2}}{3 / 2}+\frac{x^{5 / 2}}{5 / 2}\right]_{0}^{16}$

$V=2 \pi\left[2 / 3 \cdot\left(4^{2}\right)^{3 / 2}+2 / 5\left(4^{2}\right)^{5 / 2}\right]$

$V=4 \pi / 3 \cdot 4^{3}+4 \pi / 5 \cdot 4^{5}$

$V=4 \pi\left(\frac{1}{3}+\frac{4^{2}}{5}\right)$

$V=\frac{256}{15}(5+48) \pi$

$V=\frac{256 \times 53}{15} \pi$

$V=\frac{13568}{15} \pi$

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