Question

[ CALCULUS III - DOUBLE INTEGRALS (fig. inside) ] Set up an integral for both orders of integration, and use the more convenient order
to evaluate the double integral. f(x, y) = xy, R: rectangle with vertices (0, 0), (0, 5), (3, 5), (3, 0).

Answer 1

Set up an integral for both orders of integration, and use the more convenient order
to evaluate the double integral \[\int \int_R f(x,y)\ dA\] f(x, y) = xy, R: rectangle with vertices (0, 0), (0, 5), (3, 5), (3, 0).

(see attached image)

Answer 2

substitution to obtain\[\int \int_Rxy\ dA\]

Answer 3

@SithsAndGiggles

R: rectangle with vertices (0, 0), (0, 5), (3, 5), (3, 0).

Answer 4

Since x and y are relatively the same, I do not imagine there being a "more convenient" order for this one.

Answer 5

There doesn't appear to be a more convenient order.

Answer 6

But, I guess you could integrate WRT x first.

Answer 7

I got 225/4 either way. Let me know if you got something different. @stamp

Answer 8

@calmat01 I am not sure how to solve these types of problems. I have yet to read notes on double integrals relating to rectangles in R2, could you discuss with me the steps in how to set up this integral?

Answer 9

notes — http://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx

Answer 10

R: [0, 3] x [0, 5]

Answer 11

\[\int_0^5\int_0^3xydxdy\]\[\int_0^5\frac{x^2y}{2}|_0^3dy\]

Answer 12

Honestly I am going to check this on some paper, I will let you know if I get 225/4

Answer 13

ok, My java isn't working properly. All I am seeing is a bunch of commands in your post and so not sure if you have the outter limits from 0 to 5 and the inner limits from 0 to 3, but there is a theorem by Fubini that says it makes no difference which way you integrate first, you should come up with the same result either way. However, sometimes integrating first WRT one variable is much easier than WRT the other.

Answer 14

I'll be around, just holler if you get stuck.

Answer 15

\[\int_0^5\int_0^3xy\ dxdy=225/4\]

Answer 16

I got 225/4 so we are ok.

These increases in difficulty, I am currently at work but I will be posting more of these as the day goes on if you are interested in some practice. Thank you for your verification.

Answer 17

No problem. And I can always use the practice.

Answer 18

Everything seems right to me.