Different answers when switching the order of integration

[hwill205]

Homework Statement

Evaluate the following iterated integrals using the above commands. In each case check your answer by reversing the order of integration in the iterated integral if possible.

Homework Equations

As a type 1 region:

0<x<1 and x<y<2x

\int\int xy² dy dx

As a type 2 region:

0<y<2 and y/2<x<y

\int\int xy² dx dy

The Attempt at a Solution

When I integrated as a type 1 region, I got 7/15. When I integrated as a type 2 region, I got 12/5.

Are the answers supposed to be different for this particular problem? Any help is gladly appreciated.

 

[Mark44]

Homework Statement

Evaluate the following iterated integrals using the above commands. In each case check your answer by reversing the order of integration in the iterated integral if possible.

Homework Equations

As a type 1 region:

0<x<1 and x<y<2x

\int\int xy² dy dx

As a type 2 region:

0<y<2 and y/2<x<y

\int\int xy² dx dy

The Attempt at a Solution

When I integrated as a type 1 region, I got 7/15. When I integrated as a type 2 region, I got 12/5.

Are the answers supposed to be different for this particular problem? Any help is gladly appreciated.

Yes, the answers should be different. Although the integrands are the same in both problems, the regions as described are subtly different.

In the first problem, the region is the triangle bounded by y = x, y = 2x, and x = 1. The vertices of this triangle are the origin, (1, 1), and (1, 2).

In the second problem, the region is a different triangle, one whose bounds are the line x = y/2, the line x = y, and the line y = 2 (not the line x = 1). The vertices of this triangle are the origin, (1, 2), and (2, 2). Since this is a larger triangle, and the integrand is nonnegative everywhere in the first quadrant, it's to be expected that the integral would be larger.