Multivariate Calculus - Bruce E. Shapiro

132 LECTURE 16. DOUBLE INTEGRALS OVER RECTANGLES
Whenever we calculate a definite integral, the final answer will always be a number
and will not have any variables left in it.
Example 16.4 Find
∫ 2 ∫ 2
0
0
(x 2 + y 2 )dxdy
Solution.
∫ 2 ∫ 2
0
0
(x 2 + y 2 )dxdy =
=
=
=
∫ 2
0
∫ 2
0
∫ 2
0
(∫ 2
x 2 dx +
0
[
1
∫ 2
∣ ∣∣∣
2 ∫ 2
3 x3 + y 2 dx
0 0
[ ] 8
3 + 2y2 dy
( 8
3 y + 2 3 y3 )∣ ∣∣∣
2
= 8 3 (2) + 2 32
(8) =
3 3
0
0
)
y 2 dx dy
]
dy
Volumes and Areas
The Area A of a rectangle R in the xy plane is
A = dxdy
R
The Volume V between the surface z = f(x, y) and the rectangle R in the xy plane
is
V = f(x, y)dxdy
R
These formulas for area and volume still hold even if R is not a rectangle (even
though we have not yet defined the concept of an integral over a non-rectanglular
domain).
Example 16.5 Find the area of the rectangle [0, l] × [0, w].
Solution. Using double integrals,
∫ w
A =
R dxdy =
0
∫ l
0
dxdy = lw
Example 16.6 Find the volume of the solid under the plane z = 2x + 4y and over
the rectangle [3, 12] × [2, 4].
Revised December 6, 2006. Math 250, Fall 2006