Double Integral Homework: Evaluate & Change Order if Necessary

[cse63146]

Homework Statement

Evaluate the following double integral. Change order of integration if necessary.

$$\int^{1}_{0} \int^{x}_{0} x^2 sin(\Pi x y) dy dx$$

Homework Equations

The Attempt at a Solution

$$\int^{1}_{0} \int^{x}_{0} x^2 sin(\Pi x y) dy dx = -\frac{1}{\Pi}\int^{1}_{0} x cos(\Pi x^2 ) dx$$

Let u = x^2 and du = 2x dx

$$- \frac{1}{2 \Pi} \int^{1}_{0} cos (\Pi u) du = -\frac{1}{2 \Pi} \frac{sin (\Pi x^2 )}{\Pi} |^{1}_{0} = - \frac{sin( \Pi)}{2 \Pi^2} = 0$$

but that's wrong. Anyone catch my mistake?

I was also wondering when I'm supposed to change the order of integration. Thanks.

 

[AssyriaQ]

Hint: What is the value of $$\cos(0)$$?

 

[cse63146]

cos(0) = 1

I see what you meant, let me try it

 

[AssyriaQ]

I was referring to when you integrated the sine with respect to y, and received a cosine. The lower integration limit is 0 so it cos(0) should not vanish.



Edit: I see you discovered what I meant. It took 20+ minutes for this computer to load my reply!

 

[cse63146]

I was referring to when you integrated the sine with respect to y, and receive a cosine. The lower integration limit is 0 so it cos(0) should not vanish.

Took me a little while, but I got it. Thank you.

I was just wondering when the right time to change the order of integration is, since we never covered it in class.

 

[HallsofIvy]

When it is convenient. Certainly if you can't do a double integral in a given order you should try changing the order.

 

[cse63146]

When it is convenient. Certainly if you can't do a double integral in a given order you should try changing the order.

That certainly makes sense. Thank You.