Areas and lengths in polar coordinates

Double Integrals over General Regions1

Double Integrals over General Regions Type IDouble 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 = h1(y) and x = h2(y), for c y d.Double Integrals over General Regions Type II

Double Integrals over General Regions Example 2Evaluate 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 3The region D is given. Set up both ways if possible:

Type II:

Type I:

Type I:

Type II:

Double Integrals over General Regions Example 4Reverse 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 5Recall 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 x2 and the x-axis.We can look at the region D as Type I region:

Double Integrals over General Regions Example 6The 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