How do I integrate these functions?

[noobishnoob]

Hi,

I am currently working on two separate problems that involve the integration of sin(x²) and e^r2 with respect to r.

As you can see in the image, the question asks to evalute the double integral using polar coordinates.

Since r=sqrt(x²+y²), the integral becomes sin(r²) and e^r2. But how do I go ahead with this? It looks like a fairly simple integration, but in my experience it did not turn out to be so.

I have tried integrating this by parts and gone around in circles where the result has to be integrated by parts again and again. I don't seem to recognise any integration method of solving it.

Any advice would be appreciated, thanks.

 

[Steven G]

Can you please show us your work so we can see where you are having problems.
You only made some substitutions, like x²+y²but what about dxdy and the limits. Again, please show us your work and we'll help you get through these problems.
Also we really prefer one problem per post.

 

[noobishnoob]

Can you please show us your work so we can see where you are having problems.
You only made some substitutions, like x²+y²but what about dxdy and the limits. Again, please show us your work and we'll help you get through these problems.
Also we really prefer one problem per post.

Yeah well it's technically just one problem once I figure out how to do the integral cuz the two parts are pretty much the same thing.

Also here are my workings fr part (a), I dont know how to proceed from here.

 

[Dr.Peterson]

Were you taught how to convert a double integral in Cartesian coordinates to polar? Are you aware that dxdy and drdθ are not the same thing? This is what Jomo already referred to.

Please show what you learned about this, so we can help you use the method you were taught. (And if you haven't been taught it, look it up, in your book or elsewhere.)

 

[noobishnoob]

Were you taught how to convert a double integral in Cartesian coordinates to polar? Are you aware that dxdy and drdθ are not the same thing? This is what Jomo already referred to.

Please show what you learned about this, so we can help you use the method you were taught. (And if you haven't been taught it, look it up, in your book or elsewhere.)

Ah right I think I made a silly mistake.

The method I was taught is we say x=rcostheta and y=rsintheta, and to rewrite the integral in terms of these new variables. Additonally the new integral had to be multiplied by the Jacobian, and thata's what I forgot to include. I'll try and see if I can get the answer from there