Integral Calculus Answers

Find the two repeated limits at (0,0) of
f(x,y) = {(x-y)2}/(x2 + y2) , (x,y)not equal to (0,0)
=0 , (x,y) = (0,0) .
Also find the simultaneous limit , if it exists .

Find the directional derivative of
f(x,y) = (2 x y2)/(x2 + y2) , (x,y) not equal to (0,0)
=0 , (x,y) = (0,0) .
at (0,0) in the direction thita= (pi)/3 .

If f(x,y) = (4 x2 y2)/(x2 + y2) when (x,y) not equal to zero, then show that fxy(0,0)=fyx(0,0) . And
=0 when x=0=y

If f(x,y) = (4 x2 y2)/(x2 + y2) where (x,y) not equal to zero .
then show that fxy(0,0)=fyx(0,0) .

Find the domain of the function f and evaluate f(1,3) .
f(x,y)={(x+y+1)1/2} /(x-1) .

Evaluate the integral of f(x,y,z) =x+2y-z over the cylinder bounded by x2+y2=4 , z=0 and z=1 .

Show that the function F(x,y) = ((e)xy , lnx) is locally invertible at (1,3) .

Find the work done by the force F=(x2 , -xy) in moving a particle from (0,0) to (1,2) along the parabola y=2 x2 from (0,0) to (1,2) .

The density at any point on a semicircular lamina is proportional to the distance from the centre of the circle . Find the centre of gravity of the lamina .