Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = 7 x , y = 0, x = 1; about x = −4

Answer 1

Answer:

So, the volume V is

$V=\frac{14\pi}{3}$

Step-by-step explanation:

We have that:

$y=7x\\\\y=0\\\\x=1\\\\x=-4$

We have the formula:

$V=2\pi\int_a^b x(g(x)-f(x))\, dx\\\\g(x)>f(x).$

We calculate the volume V, we get

$V=2\pi\int_a^b x(g(x)-f(x))\, dx\\\\V=2\pi\int_0^1 x(7x-0)\, dx\\\\V=2\pi\int_0^1 7x^2\, dx\\\\V=2\pi\cdot 7\left[\frac{x^3}{3}\right]_0^1\\\\V=14\pi\left(\frac{1}{3}-\frac{0}{3}\right)\\\\V=\frac{14\pi}{3}$

So, the volume V is

$V=\frac{14\pi}{3}$

We use software to draw the graph.