LINE INTEGRALS- PROBLEMS & SOLUTIONS #lineintegrals #problems #solutions

 LINE INTEGRALS - PROBLEMS &SOLUTIONS :

      We have discussed about the definitions and development of line integrals. 

Here we going to solve some problems regarding line integrals.

 PROBLEM 1 : . Evaluate  ∫ dx/(x+y), Where C  is the curve x=at²,y=2at, 0≤t≤2.

SOLUTION : 

                       

PROBLEM 2: Show that ∫ xy dx =4/5, where C is the arc of the parabola y²=x from (1,-1) to   (1,1).

SOLUTION : 

                      

3.  PROBLEM 3 :  Show that ∫ [(x-y)³dx+(x-y)³dy]=3πa⁴, where C Is the circle x²+y²=a² in       the counter clockwise sense.

        SOLUTION : 

                         

         PROBLEM 4 : Find the value of ∫(x+y²)dx+(x²-y)dy, taken in the clockwise sense along     the closed curve C formed by y²=x and y=x between (0,0) and (1,1).

            SOLUTION :

                      

         PROBLEM 5:  Show that∫(xdy-ydx)/x²+y²= -2π, round the circle x²+y²=1; or any simple   closed curve containing the origin in its interior.

          SOLUTION :

                                

        PROBLEM 6 :  Evaluate ∫x²+y²dx and ∫x²+y² dy where C is the arc of the parabola y²=4ax between  (0,0) and (a,2a).

          SOLUTION :

                          

        PROBLEM 7 : Show that ∫x² dx+xy dy=0, where C is the arc of the circle x²+y²=1 from (1,0)  to (0,1).

         SOLUTION :

                                

         PROBLEM 8 : Evaluate  ∫  (2x²+y²)dx +(3y-4x)dy, Where C is the boundary of triangle ABC whose vertices are A=(0,0), B=(2,0),C=(2,1).

          SOLUTION :

                  

PROBLEM 9 : Evaluate ∫(ydx-xdy), , where C is given by the ellipse x²/a² + y²/b² = 1 in the anticlockwise direction.

     SOLUTION :

                  

       PORBLEM 10 : Evaluate ∫(3xydx-y²dy), where C is the curve In the xy-plane y=2x² from (0,0) to (1,2).

          SOLUTION :

                          

      PROBLEM 11 : Find the value of ∫x²y dx+y²x dy taken in the clock wise sense along the hexagon whose vertices  are (∓3a,0),( ∓2a, ∓3a).

       SOLUTION :

                 

                       

      

                 

             

PROBLEM 12  :  Show that ∫(x²dy-y²dx)/(x^5/3+y^5/3) = (3π/16) a^4/3, where C is the quarter of the asteroid x= a cos³t, y= a sin³t, from (a,0) to (0,a).

 SOLUTION :

                  

PROBLEM 13: Evaluate  ∫(x²+y²)dx-2xydy, where C is the rectangle x=0,x=a,y=0,y=b.

 SOLUTION :

                        

PROBLEM  14 : Evaluate ∫ (2a-y)dx-(a-y)dy, where C is given by x=a(t-sint), y= a (1-cost).

SOLUTION :