# Order Of Integration Switch

• Nov 6th 2011, 12:46 PM
spruancejr
Order Of Integration Switch
I've been given the double integral $\int_{x= 0}^{1}\int_{y=x}^{3x}xy dydx$ and instructed to change the order of integration and solve. After solving the given integrals I found they equaled 1.

I reasoned that the initial graph is bounded by two lines y=x and y=3x and bounded on the x-axis between x=0 and x=1; this formed a skewed sort of triangle.

Here's what I changed it to: $\int_{y= 0}^{3}\int_{x=(1/3)y}^{x}xy dxdy$

I either I'm I'm messing up or my professor's solution is wrong...help?
• Nov 6th 2011, 02:01 PM
TKHunny
Re: Order Of Integration Switch
Did you just change around all the x's and y's? You must actually look at the geometry.

First Piece $\int_{0}^{1}\int_{y/3}^{y} xy\;dxdy$

Second Piece $\int_{1}^{3}\int_{y/3}^{1} xy\;dxdy$
• Nov 6th 2011, 02:24 PM
spruancejr
Re: Order Of Integration Switch
When I solve the two integrals, I get 9 as opposed to 1 which was the answer prior to switching the order of integration. Should the two answers be the same?
• Nov 6th 2011, 02:47 PM
Plato
Re: Order Of Integration Switch
Quote:

Originally Posted by spruancejr
When I solve the two integrals, I get 9 as opposed to 1 which was the answer prior to switching the order of integration. Should the two answers be the same?

Check your upper bound on the second set of integrals.
$\int_1^3 {\int_{y/3}^1 {xydxdy} }$
• Nov 6th 2011, 03:51 PM
spruancejr
Re: Order Of Integration Switch
I'm confused by the second integral going from 0 to 1. Shouldn't it be from 1 to 3 since the second (top triangle) is bounded between 1 and 3 in the y direction?
• Nov 6th 2011, 03:56 PM
Plato
Re: Order Of Integration Switch
Quote:

Originally Posted by spruancejr
I'm confused by the second integral going from 0 to 1. Shouldn't it be from 1 to 3 since the second (top triangle) is bounded between 1 and 3 in the y direction?

CORRECTED NOW.
That was a copy&paste error.