Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 7x6, y = 7x, x ≥ 0; about the x-axis

Answer 1

Answer:

The volume is $\frac{490\pi}{39}$ cubic units.

Step-by-step explanation:

The given curve is

$y=7x^6$

The given line is

$y=7x$

Equate both the functions to find the intersection point of both line and curve.

$7x^6=7x$

$7x^6-7x=0$

$7x^6-7x=0$

$7x(x^5-1)=0$

$7x=0\rightarrow x=0$

$x^5-1=0\rightarrow x=1$

According to washer method:

$V=\pi \int_{a}^{b}[f(x)^2-g(x)^2]dx$

Using washer method, where a=0 and b=1, we get

$V=\pi \int_{0}^{1}[(7x)^2-(7x^6)^2]dx$

$V=\pi \int_{0}^{1}[49x^2-49x^{12}]dx$

$V=49\pi \int_{0}^{1}[x^2-x^{12}]dx$

$V=49\pi [\frac{x^3}{3}-\frac{x^{13}}{13}]_0^1$

$V=49\pi [\frac{1^3}{3}-\frac{1^{13}}{13}-(0-0)]$

$V=49\pi [\frac{1}{3}-\frac{1}{13}]$

$V=49\pi (\frac{13-3}{39})$

$V=49\pi (\frac{10}{39})$

$V=\frac{490\pi}{39}$

Therefore the volume is $\frac{490\pi}{39}$ cubic units.