Multiple Integration

Example

Let's evaluate the double integral R6xydA , where R is the region bounded by y=0, x=2, andy=x2. We will verify here that the order of integration is unimportant:

Integrating first with respect to y, then with respect to x: R6xydA = = = = = = 020x26xydydx 023xy2   x2y=0dx 023x5dx 21x6     2x=0 21(64)−21(0) 32

Integrating first with respect to x, then with respect to y: R6xydA = = = = = = 042y6xydxdy 043x2y   2x=ydy 0412y−3y2dy 6y2−y3   4y=0 6(4)2−(4)3−6(0)2−(0)3 32

so R6xydA=32  here, regardless of the order in which we carry out the integration, as long as we are careful to set up the limits of integration correctly.

Now for a triple integral...