Question

Consider the region bounded by the curves 2. y = 3 + x , y = 3, and x = 2. Find the volume of the solid obtained by rotating this region about the axis y = 3.

Answer 1

Answer:

$\frac{8\pi}{3}$ units cubed

Step-by-step explanation:

Let's look at a point (x,y) on the line y=3+x. The height between (x,y) and the x-axis is y.  We want the distance from the axis of rotation which is y=3 so the height (or distance between) point (x,y) on y=3+x and y=3 is y-3.  

y-3 is the radius in terms of y.

(3+x)-3=x is the radius in terms of x.  I replaced y with 3+x since we have y=3+x.

The area of the circle I drew is $\pi \cdot r^2=\pi \cdot x^2$

To find the volume we must integrate the area of the circle we found between the bounded lines x=0 and x=2.

$\int_0^2 \pi \cdot x^2$

$\pi \cdot \frac{x^3}{3}|_0^2$

$\pi[\frac{2^3}{3}-\frac{0^3}{3}]$

$\pi[\frac{8}{3}-0]$

$\pi[\frac{8}{3}]$

$\frac{8\pi}{3}$ units cubed