Question

Evaluate the following integrals by using Polar coordinates.

   (a)∬Rcos(x2+y2)dA, where R={(x,y)∈R2∣x2+y2<4}

 

 

Answer 1

Solution

Evaluate the following integrals by using Polar coordinates.

   (a)∬Rcos(x2+y2)dA, where R={(x,y)∈R2∣x2+y2<4}

                     

 Let{x=rcosθy=rsinθ⇒x2+y2=r^2

we  have∬Rcos(x2+y2)dA=∫∫R′cos(r2)rdrdθ

where  A′={(r,θ)∈R2:0≤r≤2,0≤θ≤2π}

Then∬Rcos(x2+y2)dA=∫2π0∫20cos(r2)rdrdθ=

  ∫=∫2π012sin(4)dθ=πsin(4)

ysin(4)