# Computing Double Integrals and Volumes with Maple

## Exercises on Double Integrals

Iterated integrals are straight forward in Maple. Here are some examples,

> Int(Int(x^4-y^2,x=1..2),y=0..1) = int(int(x^4-y^2,x=1..2),y=0..1);

```                            1   2
/   /
|   |   4    2          88
|   |  x  - y  dx dy = ----
|   |                   15
/   /
0   1```
> Int(Int(sin(x+y),x=0..Pi/2),y=0..Pi/2)=
> Int(int(sin(x+y),x=0..Pi/2),y=0..Pi/2);
```          1/2 Pi  1/2 Pi                     1/2 Pi
/       /                          /
|       |                          |
|       |     sin(x + y) dx dy =   |     sin(y) + cos(y) dy
|       |                          |
/       /                          /
0       0                          0```
 and this last integral is equal to:

> Int(int(sin(x+y),x=0..Pi/2),y=0..Pi/2) =
> int(int(sin(x+y),x=0..Pi/2),y=0..Pi/2);

```                          1/2 Pi
/
|
|     sin(y) + cos(y) dy = 2
|
/
0```
 Double integrals can be used to compute volumes. Here are some examples.

### Problem1:

 Find the volume of the solid lying under the elliptic paraboloid

> z := 1 - (x^2/4 + y^2/9);

```                                          2        2
z := 1 - 1/4 x  - 1/9 y```
 and above the square R=[-1,1]x[-2,2]. The solid looks like this:

> plot3d(z,x=-1..1,y=-2..2,axes=frame);

#### Solution1:

 Computing the integral over the square we get:

> Int(Int(z,x=-1..1),y=-2..2) = int(int(z,x=-1..1),y=-2..2);

```                      2   1
/   /
|   |           2        2         166
|   |  1 - 1/4 x  - 1/9 y  dx dy = ---
|   |                               27
/   /
-2  -1```
 which is approximately,

> ans1 := evalf(int(int(z,x=-1..1),y=-2..2),2);

`                                  ans1 := 6.1`

### Problem2:

 Find the volume of the solid in the first octant bounded by the surface,

> z2 := x*sqrt(x^2+y);

```                                        2     1/2
z2 := x (x  + y)```
 and the planes x=1, and y=1.

#### Solution2:

 The double integral will compute the volume as the limit of approximations with blocks like the ones shown in the picture below:

> with(mvcal):
> blockapp(z2,x=0..1,y=0..1);

 The actual surface on the first octant looks like this,

> plot3d(z2,x=-1..2,y=-1..2,axes=frame);

 Computing the double integral we get,

> Int(Int(z2,x=0..1),y=0..1);

```                            1   1
/   /
|   |      2     1/2
|   |  x (x  + y)    dx dy
|   |
/   /
0   0```
 actually maple had a hard time with this integral in my computer. It refused to compute the exact expression for the integral eventhough a simple substitution (u=x2) does it. Any way after a lot of computing...

> evalf(");

`                                  .4875805666`
 If we do the integration over x by hand and let maple handle the integration over y we get the exact answer as,

> ans2 := int((1+y)^(3/2)-y^(3/2),y=0..1)/3;

```                                          1/2
ans2 := 8/15 2    - 4/15```
> evalf(");
`                                  .4875805663`

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>