Thread: Finding volume with multiple integrals.

1. Finding volume with multiple integrals.

Could someone explain to me how to solve the following problem? I'm not sure how to do it.

Find the volume of the solid enclosed by the graphs of the given equations:

z=x^2+4
y= 4-x
^2
x+y=2
z=0

I think one of the limits should be z=0 to z=x
^2+4, but I'm not sure what would be the other limit. Also I'm not sure what is the function that I have to integrate. Please guide me, folks. Thanks in advance.

2. Re: Finding volume with multiple integrals.

Since you are just finding a volume and not integrating a function over a volume the integral will be

$\int \int \int dxdydz$ with the limits added

You can do the integration in any order you like, I think the easiest is z then y then x

$\int \int \int dzdydx$

As you rightly pointed out, the limits on the z integral are 0 to 4+x2

The limits on the y integral are got from the equations y= 4-x2 and x+y=2 which will be y= 4-x2 and y=2-x. Now to decide which is the lower and which is the upper limit, the lower limit will have the smaller value. 4-x2 is always less than or equal to 2-x so 2-x should be the upper limit.

The limits on the x integral are also got from the equations y= 4-x2 and y=2-x. Putting them equal to each other you find that 4-x2=2-x The solutions to that equations are the limits for x.

3. Re: Finding volume with multiple integrals.

Originally Posted by Shakarri
Since you are just finding a volume and not integrating a function over a volume the integral will be

$\int \int \int dxdydz$ with the limits added

You can do the integration in any order you like, I think the easiest is z then y then x

$\int \int \int dzdydx$

As you rightly pointed out, the limits on the z integral are 0 to 4+x2

The limits on the y integral are got from the equations y= 4-x2 and x+y=2 which will be y= 4-x2 and y=2-x. Now to decide which is the lower and which is the upper limit, the lower limit will have the smaller value. 4-x2 is always less than or equal to 2-x so 2-x should be the upper limit.

The limits on the x integral are also got from the equations y= 4-x2 and y=2-x. Putting them equal to each other you find that 4-x2=2-x The solutions to that equations are the limits for x.
Thank you so much! That was very helpful.