Calculus Answers

Evaluate D∫∫(x+2y) da, where D is the region bounded by the parabolas y=2x^2 and y= 1+x^2.

Check if the following integrals are independent of path and evaluate those which are independent. i) ∫ (0,0) to (3,4) (6xy-y^3)DX +(3x^2-x^3y)dy. 2) integration (-1,4) to (3,8) (3x^2-2y^2)dx-4xy dy

Find the moment of inertia I2 for the solid above the xy-plane bounded by the paraboloid z=x^2+y^2 and the cylinder x^2+y^2 =9 assuming the mean density to be constant C.

A torus (a dough nut-shaped object) is formed by revolving the circle

x^2 + y^2 = a^2

about the vertical line

x = b,

where 0 < a < b. Find its volume.

Find the centre of gravity of a mass in the shape of a semicircular disc of radius 4, if the density at (x,y) is 2y/x^2+y^2.

Write ∫0 to 1∫0 to √1-x^2(√1-y^2) Dy DX as an integral over a region D. Sketch the region D and show that it is of both types 1 and 2. Reverse the order of integration and evaluate it.

Prove that the functions g(x,y) = 2x -3y / 4x+5y and h(x,y) = x/y , y≠0,y ≠ -4/5 ×x are functionally dependent.

Find the values of a and b for which the following limit exists. lim x→0 ae^x+e^-x+bx/1-cosx

Apply Inverse function theorem to check the local invertibility of the following function f: R^2→R^2 given by at the point (0,π). F(x,y) =(ycosx,x-y+2)

Evaluate: lim x→infinity(√2x^2+3x-2 -√2x^2-3x+2)