# Calculus I homework question?

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. and sketch the region

y = 7sqrt(64 − x^2), y = 0, x = 1, x = 3;

V = ___ about the x-axis Relevance

This might help you to see how to set up the integral.

An element of volume is created by rotating a rectangle y*dx around x axis.

That volume element will be coin shaped looked at edge on, although usually called a disc, and should be seen as a stubby cylinder with volume like πr^2*h.

πr^2 = πy^2, and the height/depth h is taken as dx for an element of volume.

Since y^2 = 49(64 – x^2), the integrand becomes 49π(64 – x^2)dx

The x limits go from 1 to 3, so you get

V = {x: 1 to 3} 49π∫(64 – x^2)dx

You should find ∫(64 – x^2)dx quite easy

it is 64x – x^3/3, so applying those limits

V = 49π[192 – 9 – (64 – 1/3)] = 49π *359/3 = 17542π/3 ~ 18369.9

• hey4 months agoReport

Thanks so much!

• V =

3

∫ πy^2 dx = ∫π*49(64-x^2 dx [1,3]

1

= π[3136x-49x^3/3] [1,3]

= π([3136*3-9*49] - [3136-49/3])

= π(8967-3136+49/3)

17542π/3

• hey4 months agoReport

Thanks so much!

• Have you not been taught the method of disks and washers?

The area of some disk is pi [ (7sqrt(64 − x^2))^2 - 0^2 ] so that's what we will integrate

pi * int[1 to 3] 49(64-x^2) dx

= ?