Question

Set up the integral that uses the method of disks/washers to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified lines.
y=3\sqrt(x), y = 3, x= 0

a.) about the line y = 3

b.) about the line x = 5

Answer 1

Answer:

$a)\: \frac{3}{2} \pi$

$b) \:\frac{47}{5} \pi$

Step-by-step explanation:

$y=3\sqrt x,\:\:y=3,\:\:x=0$

a) about the line y = 3

$3\sqrt x=3$  ⇒  $x = 1$  is the intersection point

So,

$V=\int\limits^1_0\pi(3-3\sqrt x)^2\:dx=\pi\int\limits^1_0(9-18\sqrt x+9x)\:dx=\\\\=\pi(9x-12x^{3/2}+9/2x^2)|^1_0=\pi(9-12+9/2)=\frac{3}{2} \pi$

b)  about the line x = 5

$y=3\sqrt x$  ⇒  $x=y^2/9$

So,

$V = \int\limits^3_0\pi([5-0]^2-[5-y^2/9]^2)\:dy=\pi\int\limits^3_0(25-25+10y^2/9-y^4/81)\:dy=\\\\=\pi(10y^3/27-y^5/405)|^3_0=\pi(10-3/5)=\frac{47}{5} \pi$