volume by cylindrical shells

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=17 sqrt(x), y=0, x=1; about x=-2

Answer 1

Hi there!

We want to solve for the volume using the given shell. First, we'll need to understand the 2-D shell that we're rotating.

We're given a horizontal boundary of y=0, and a vertical boundary of x=1. Let's make these boundaries consistent to x. We can find that 17√x = 0 when x = 0, of course, so now we know that our boundaries are from a=0 to b=1 on the x-axis. Likewise, we know that for x>0, 17√x remains positive.

The method of cylindrical shells can be used to find volume when rotated around a vertical axis using the following equation:

V = ⌠b C [f(x) - g(x)] dx

⌡a

Breaking this down, V represents volume, C represents circumference, and f(x) - g(x) integrated with respect to x between a and b represents the area (that one was a mouthful). In a physical sense, this equation gives us our volume by adding together each circumference multiplied by each corresponding area for every value of x within our bounds.

To fit this equation to our problem,

C = 2π (x - (-2))

= 2π (x+2)

f(x) = 17√x

g(x) = 0 because one of our bounding functions is y = 0

a = 0 (we found these two earlier)

b = 1

So now, we plug it all into our cylinder volume equation above!

V = 2π ⌠b (x+2) [17√x - 0] dx ; where a = 0 and b = 1.

⌡a

= 2π ⌠b 17x^1.5 + 34√x dx

⌡a

= 2π 17/2.5 x^2.5 + 34/1.5x^1.5 |b ; We integrate, with thankfully easy bounds.

|a

= 2π (17/2.5 * 1 + 34/1.5 * 1 - (0 + 0))

= 185.14

So there you have it! Be sure to double-check and let me know if you have any further questions.