Multivariate Calculus - Bruce E. Shapiro

140 LECTURE 17. DOUBLE INTEGRALS OVER GENERAL REGIONS
Example 17.5 Find the iterated integral
∫ 12 ∫ x+3
−2 −x 2
(x + 3y)dxdy
Solution. Separating the integral into two parts and factoring what we can gives
∫ 12 ∫ x+3
−2
(x + 3y)dydx =
=
=
=
=
∫ 12 ∫ x+3
−2
∫ 12
−2
∫ 12
−2
∫ 12
−2
∫ 12
−2
x
−x 2 xdydx +
∫ x+3
−x 2 dydx + 3
x(y| x+3
−x 2 )dx + 3
∫ 12 ∫ x+3
−2 −x
∫ 2
12 ∫ x+3
−2
∫ 12
−2
x(x + 3 + x 2 )dx + 3 2
(x 2 + 3x + x 3 )dx + 3 2
−x 2
3ydydx
ydydx
( 1 2 y2 | x+3 )dx
−x 2
∫ 12
−2
∫ 12
−2
[(x + 3) 2 − (−x 2 ) 2 ]dx
(x + 3) 2 dx − 3 2
∫ 12
−2
x 4 dx
Integrating and plugging in numbers, we find that
∫ 12 ∫ x+3
−2
−x 2 (x + 3y)dydx = ( 1 3 x3 + 3 2 x2 + 1 4 x4 )| 12
−2 + 3 1
2 3 (x + 3)3 | 12
−2 − 3 2
1
5 x5 | 12
= 1 3 (12)3 + 3 2 (12)2 + 1 4 (12)4 − 1 3 (−8) − 3 2 (4) − 1 2 (16)
+ 1 2 (153 − (−8)) − 3 10 (125 − (−32))
= 576 + 216 + 5184 + 8 − 6 − 8 + 1691.5 − 124432
3
≈
−116775.83
−2
Figure 17.7: The plane z = 6 − 2x − 3y and its cross-section on the xy plane.
Example 17.6 Find the volume of the tetrahedron bounded by the coordinate planes
and the plane z = 6 − 2x − 3y.
Revised December 6, 2006. Math 250, Fall 2006