Show that ∫ xy2 dy- x2y dx = 35 a4π/16 where C is the counter clockwise curve of the cardiod r=a(1+cosθ ). DOUBLE INTEGRALS #double #integrals

PROBLEM :

      Show that  ∫xy² dy- x²y dx = 35 a⁴π/16 where C is the counter clockwise curve of the cardiod 

       r=a(1+cosθ ).

SOLUTION :

 Evaluate ∬ f(x,y) dxdy where f(x,y)= (2y-1)/ x+1 , and E is the region bounded by x=0,y=0, y=2x-        4.

 *   Change the order of integration and hence show that ∬dx dy/[ ( 1+e^y) √(1-x^2-y^2)]   = (𝝿 /2)                    log(2e/(1+e) where x-0 to x=1 and y=0 to y=√(1-x^2)

 *   In the integral ∬ (4-y) dydx, change the order of integration and evaluate the integral where x=2 to            x=4 and y=4/x to y=(20-4x)/(8-x)

 *  Sketch the region of integration and evaluate ∬ xsiny dydx where x=0 to x=𝛑 and y-0 to y=x.

 *  Sketch the region of integration and write an equivalent double integral with the order of integration

     reversed for ∬ 3y dxdy where y=0 to y=1 and x= -√(1-y^2) to y=√(1-y ^2).

  *Evaluate ∬ √(4x^2-y^2) dxdy where E is the region bounded by the lines y=0,y=x and            x=1.

 *  Evaluate ∬xy(x+y) dxdy where E is the region bounded by y=x^2 and y=x.

  *  Evaluate  ∬f(x,y)dxdy , where f(x,y)=x²+y² and E={(x,y)/y=x²,x=2,y=1}   

 *  Evaluate ∬ xydxdy where E is the region bounded by xy=1,y=0,y=x,x=2.      

 * Evaluate ∬xy(x+y)² /(x²+y²) dxdy where E is region bounded y=0,y=x,x²+y²=a²                        in the first quadrant.

  *    Evaluate ∬ e^(x²+y²) dxdy , where E is the semi circular region bounded by the X-axis and                               the curve y=√(1-x²).

 *     Using polar coordinates, show that ∫dx ∫√X²+y²    dy =1/6[√2 + log ( 1+√2) ]