(873, #9) Evaluate the triple integral

where E lies under the plane z = x + 2 y and above the region in the xy-plane bounded by the curves y = x², y = 0, and x = 1.

Solution: The solid E is defined in terms of the inequalities

The latter two inequalities, 0 <= y <= x² and 0 <= x <= 1, define the base of the solid,which is displayed below at the left. For each fixed (x, y) in this region z varies from0 to x + 2 y . The domain of integration is outlined at the right.

Thus

The value of this integral is 5/28:

lprint(int( y,z));
lprint(int( y,z = 0..x + 2*y));
y*z
x*y+2*y^2

lprint(int(int( y,z = 0..x + 2*y),y));
lprint(int(int( y,z = 0..x + 2*y),y = 0 .. x^2));
1/2*x*y^2+2/3*y^3
1/2*x^5+2/3*x^6

lprint(int(int(int( y,z = 0..x + 2*y),y = 0 .. x^2), x));
lprint(int(int(int( y,z = 0..x + 2*y),y = 0 .. x^2), x = 0..1));

1/12*x^6+2/21*x^7
5/28

Maple Code

by: gm