Double Integrals over General Regions

Double Integrals over General Regions Type I Double integrals over general regions are evaluated as iterated integrals. Most double integrals fall into two categories determined by the region of integration D.

Double Integrals over General Regions Example 1

Type II Region: The left and right boundaries are graphs of continuous functions x = h 1 (y) and x = h 2 (y), for c ≤ y ≤ d. Double Integrals over General Regions Type II

Double Integrals over General Regions Example 2 Evaluate the integral where D is the region bounded by the curves y =1, y = x and y = − x +10. The given lines intersect at (1, 1), (5, 5) and (9, 1)

Double Integrals over General Regions Example 3 The region D is given. Set up both ways if possible: Type II:Type I: Type II:

Double Integrals over General Regions Example 4 Reverse the order of integration and evaluate the integral: Convert to Type I region: The integral is set up as Type II. As is, it is impossible to evaluate the integral.

Double Integrals over General Regions Example 5 Recall that represents the volume of the solid below f and above the region D. Example 5: Calculate the volume under the plane z = y and above the region bounded by y = 9 − x 2 and the x -axis. We can look at the region D as Type I region:

Double Integrals over General Regions Example 6 The two surfaces intersect along a curve C: The circle of radius 3 is the projection of C on the xy -plane and it is also the boundary of the region of integration D.

Double Integrals over General Regions Example 6 continued The limits for D, as a type I region, are: (using trigonometric substitution) Note: By symmetry of both the domain and the integrand, we can write