Suppose you want to integrate f(x,y) = 8xy over the domain you just found

Question

okay so the previous question was finding the domain ie the polar cords from

integral sign (b=5, a =0)integral sign (b=sqrt(25-x^2) a= 0) f(x,y) dy dx

found to be 0 <= r <= 5 and 0<= theta <= Pi/2

Now the next question is to do with mathmatica and can be solved without mathematica.

Suppose you want to integrate f(x,y)=8xy over the domain you just found in the previous question. In Mathematica, try:

Integrate[Integrate[8r*Cos[t]*r*Sin[t]*___, { r,0,5}],{t,0,Pi/2}]

t = theta

I can't figure out what number to multiply Sin[t} by.

also if the coding looks bad look at it like this

integral sign (b=Pi/2, a=0) integral sign (b=5, a=0) 8r * cos(t) * r*sin(t)*___

Answer 1

The area element in Cartesian coordinates is dA = dydx

In Polar coordinates it becomes dA = r drdθ

The Mathematica Integrate command assumes the "drdθ" part so you don't need to write it, but you are missing the extra r

The integral in Cartesian or Polar coordinates is equal to 625.