# Thread: Finding the area and volume of a region bounded by two curves

1. ## Finding the area and volume of a region bounded by two curves

Hi all,

I am a bit confused as to how to find the region R bounded by two graphs.

The specific problem is:

The region R shown is bounded by the graphs of y=2^-x and y=2cos(x).

a) Find the area of R
b) Determine the volume of the solid generated by revolving R about the line y=3
c) Determine the volume of the solid generated by revolving R about the line y=-1
d) Determine the volume of the solid whose base is region R and cross sections parallel to the y-axis are squares.

2. ## Re: Finding the area and volume of a region bounded by two curves

You need to post some of your own OR tell us what kind of help you need.
BUT remember that this is not a homework service.

3. ## Re: Finding the area and volume of a region bounded by two curves

Thank you for your reply, Plato. I understand that this isn't a homework service; I wish to help others as best I can and share my knowledge while also learning new concepts.

In terms of the specific help that I need, I want to better understand the relationship between the area and volume of a region between two curves. Is the volume of a region between two curves simply the integral of its area?

4. ## Re: Finding the area and volume of a region bounded by two curves

Originally Posted by Yawnee
Thank you for your reply, Plato. I understand that this isn't a homework service; I wish to help others as best I can and share my knowledge while also learning new concepts.

In terms of the specific help that I need, I want to better understand the relationship between the area and volume of a region between two curves. Is the volume of a region between two curves simply the integral of its area?
A volume does not have area. You can integrate over two-dimensional slices of the volume, and each two dimensional slice has area. Is that what you mean?

5. ## Re: Finding the area and volume of a region bounded by two curves

Yes, that's what I'm referring to! I should have specified.

6. ## Re: Finding the area and volume of a region bounded by two curves

To find the area of the region bound between two curves, integrate the "upper" curve. This will give you the area under the upper curve. Subtract the integral of the "lower" curve. This will take away the area under the lower curve. What is left is the area between the two curves. So you will need to know where the two curves meet. They will intersect an infinite number of times, but $R$ is specifically bounded by two points where they meet.

7. ## Re: Finding the area and volume of a region bounded by two curves

Originally Posted by SlipEternal
To find the area of the region bound between two curves, integrate the "upper" curve. This will give you the area under the upper curve. Subtract the integral of the "lower" curve. This will take away the area under the lower curve. What is left is the area between the two curves. So you will need to know where the two curves meet. They will intersect an infinite number of times, but $R$ is specifically bounded by two points where they meet.
The real problem is that the points of intersection of the two curves have awful values.
Something like $x\approx~-0.659957715112726\cdots\text{ to }x\approx~1.3770899215870\cdots$.
This a one of the poorest questions I have ever seen.

8. ## Re: Finding the area and volume of a region bounded by two curves

Thank you kindly, that makes sense. I appreciate the help.

9. ## Re: Finding the area and volume of a region bounded by two curves

SlipEternal- Thank you kindly, that makes sense. I appreciate the help.

Plato- That, I believe, was what got me so immensely confused in the first place. The values give the impression that you've done something wrong when you haven't.