Iterated Integral: Changing the Order of Integration

[s10dude04]

Homework Statement

(a) For the iterated integral \\cos(x/y)dydx (inner limits x to 1, outer limits 0 to 1) sketch the region in the plane corresponding to the double integral this interated integral evaluates.

(b) Evaluate the double integral by changing the order of integration in the iterated integral and evaluating the resulting iterated integral.

Homework Equations

The Attempt at a Solution

Taking the integral of cos(x/y) with respect to y gives you sin(2x/y^2) no?

Then you evaluate from x to 1, which would be sin(2x)-sin(2/x)?

Then doing the integral of that gives ********?

Am I heading in the right direction?

 

[Mark44]

Homework Statement

(a) For the iterated integral \\cos(x/y)dydx (inner limits x to 1, outer limits 0 to 1) sketch the region in the plane corresponding to the double integral this interated integral evaluates.

(b) Evaluate the double integral by changing the order of integration in the iterated integral and evaluating the resulting iterated integral.

Homework Equations

The Attempt at a Solution

Taking the integral of cos(x/y) with respect to y gives you sin(2x/y^2) no?

Then you evaluate from x to 1, which would be sin(2x)-sin(2/x)?

Then doing the integral of that gives ********?

Am I heading in the right direction?

Not at all, as far as I can see from your work. The first part asks you to sketch the region over which integration takes place. Have you done that? This region can be described pretty simply.
For the second part, how to you go from $\int cos(x/y)dy$ to sin(2x/y^2)? If you check this work by differentiating sin(2x/y^2) with respect to y, do you get cos(x/y)?