You are here:

Calculus/volume integration(from pool)

Advertisement

Expert: - 2/10/2010


Question
1) A ball of radius 16 has a round hole of radius 9 drilled through its center. Find the volume of the resulting solid.

For this problem I'm not sure how I set up the integral to get the answer. I'm having trouble finding an example to understand what I do. When I draw it I can visualize what it looks like but I don't know where I go from there.

can you help me set up the integral?

Answer

Ring
Questioner:    Ray
Question: 1) A ball of radius 16 has a round hole of radius 9 drilled through its center. Find the volume of the resulting solid.

For this problem I'm not sure how I set up the integral to get the answer. I'm having trouble finding an example to understand what I do. When I draw it I can visualize what it looks like but I don't know where I go from there.

can you help me set up the integral?
.................................
X^2 + 9^2 = 16^2

X = SQRT(256 - 81) = SQRT(175) = 5 sqrt(7)

Volume element is a disk or ring:

ring has inner radius  9
ring has outer radius  sqrt(16^2 - x^2)

dV = pi((16^2 - x^2) - 81) dx

Integrate:  

{5sqrt(7)
| 2 * pi((16^2 - x^2) - 81) dx
}0

That should do it.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

©2012 About.com, a part of The New York Times Company. All rights reserved.