Volume of a substance (in a circle)

[ozone]

Hi I just want to check that I am doing this math problem correctly. We were given a density function for review for an upcoming math test. We were then asked to find the volume in a circular radius given this density function. I will post the exact problem now then explain the steps I went through to solve it.
θ
The density of a crude oil in a circular slick on the surface of the ocean κ miles from the center is given by δ = 6400e^(-κ/4) gallons per square mile. Find the total amount of oil within 5 miles of the center of the slick

I decided to integrate the function with the equation Area = 2(pi)(y)(x)

Obviously I picked our density function for y and just plain x for x. The integration was slightly complex but I could handle that. I just want to make sure I picked the right function.

Thank you.

 

[LCKurtz]

Hi I just want to check that I am doing this math problem correctly. We were given a density function for review for an upcoming math test. We were then asked to find the volume in a circular radius given this density function. I will post the exact problem now then explain the steps I went through to solve it.
θ
The density of a crude oil in a circular slick on the surface of the ocean κ miles from the center is given by δ = 6400e^(-κ/4) gallons per square mile. Find the total amount of oil within 5 miles of the center of the slick

I decided to integrate the function with the equation Area = 2(pi)(y)(x)

Obviously I picked our density function for y and just plain x for x. The integration was slightly complex but I could handle that. I just want to make sure I picked the right function.
Thank you.

You didn't. The density depends on the distance from the center. If the center is (0,0) then your $k =\sqrt{x^2+y^2}$. So your integral should look like
$$\iint_R 6400e^{-\sqrt{x^2+y^2}}\, dydx$$ where R is your circular disk. You would want to set that up in polar coordinates to work it.