Multiple Integrals and Polar Coordinates

Evaluate ∫∫ e x2 + y2 dA, where R is the unit circle centered R at the origin. Solution: To integrate the exponential function, we need to get the derivative of its power. In our case we have not that so we shall change to ______ coordinate.

Multiple Integrals and their applications ______ Coordinates & r 2 = x2 + y2 x = r cos θ & y = r sin θ d A = r d r dθ P (x , y ) r θ Dr. Mohamed Abdelhakem y x

The area is a unite circle at the origin. So, r change from zero to 1 (form the origin to the radius) and θ will rotate to scan the circle i.e. 0 ≤ θ ≤ 2 π. Then, ∫∫ e x2 + y2 dA = 2π 1 ∫ ∫e 0 R 1 = 2 Dr. Mohamed Abdelhakem r2 rd rdθ 0 2π ∫ 0 1 r2   ∫ e (2 r ) d r  d θ   0 

∫∫ e R x2 + y2 1 dA = 2 2π 1 = 2 2π ∫ 0 e   r2    1 0 dθ ∫ (e − 1) d θ 0 1 2π = (e − 1) (θ ) 0 = π (e − 1) 2 Dr.

Basic Science Department Mathematics 3 BAS 111 – BSC 111 3 Credit Hrs Lecture 9 Dr. Mohamed Abdelhakem Page 1 of 41 Mathematics 3 (BAS 111 – BSC 111) Fall 2023 Lec 9 Basic Science Department Multiple Integrals and their applications ______ Coordinates & r 2 = x2 + y2 x = r cos θ & y = r sin θ d A = r d r dθ P (x , y ) r θ Dr. Mohamed Abdelhakem y x