devotion & mathematics

             Integration over Rectangular Regions                                                                Integration over Non-Rectangular Bounded Regions                                                                                                     Repeated integrals                                                                               ...

                                Double Integrals                                                      Problems & Solutions                                                               Back #vector #calculus #vector calculus

  PROBLEM :                       Using polar coordinates, show that ∫dx ∫√X 2 +y 2     dy =1/6[√2 + log ( 1+√2) ] 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 th...

PROBLEM :                          Evaluate ∬ e^(x 2 +y 2 ) dxdy , where E is the semi circular region bounded by the X-axis and                               the curve y= √ (1-x 2 ). 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 ∬ xsi...

PROBLEM :                          Evaluate ∬ xy(x+y) 2 /(x 2 +y 2 ) dxdy where E is region bounded y=0,y=x,x 2 +y 2 =a 2                         in the first quadrant.  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...

PROBLEM :       Show that  ∫ xy 2 dy- x 2 y dx = 35 a 4 π /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 ...

 PROBLEM :      Evaluate  ∬f(x,y)dxdy , where f(x,y)=x 2 +y 2 and E={(x,y)/y=x 2 ,x=2,y=1} 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 ...

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