Questions from General Calculus

Q: (a). Evaluate ∭E dV, where E is the

(a). Evaluate ∭E dV, where E is the solid enclosed by the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1. Use the transformation x = au, y = bv, z = cw. (b). The earth is not a perfect sphere; rotation has resul...

Q: An important problem in thermodynamics is to find the work done by

An important problem in thermodynamics is to find the work done by an ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is...

Q: A rectangle with length L and width W is cut into four

A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller...

Q: Evaluate the integral by making an appropriate change of variables.

Evaluate the integral by making an appropriate change of variables. ∬R (x-2y)/(3x-y) dA, where R is the parallelogram enclosed by the lines x - 2y = 0, x - 2y = 4, 3x - y = 1, and 3x - y = 8

Q: Evaluate the integral by making an appropriate change of variables.

Evaluate the integral by making an appropriate change of variables. ∬R (x + y)e^(x^2-y^2 ) dA, where R is the rectangle enclosed by the lines x - y = 0, x - y = 2, x + y = 0, and x + y = 3

Q: Use cylindrical coordinates. (a). Find the volume of

Use cylindrical coordinates. (a). Find the volume of the region E that lies between the paraboloid z = 24 - x2 - y2 and the cone z = 2 √(x^2 + y^2 ). (b). Find the centroid of E (the center of mass i...

Q: Use cylindrical coordinates. (a). Find the volume of

Use cylindrical coordinates. (a). Find the volume of the solid that the cylinder r = a cos  cuts out of the sphere of radius a centered at the origin. (b). Illustrate the solid of part (a) by graphin...

Q: Evaluate the integral by making an appropriate change of variables.

Evaluate the integral by making an appropriate change of variables. ∬R ex+y dA, where R is given by the inequality |x | + |y | < 1

Q: (a). Verify that f (x, y) =

(a). Verify that f (x, y) = {_0^4xy if 0 < x < 1, 0 < y < 1 otherwise is a joint density function. (b). If X and Y are random variables whose joint density function is the function f in part (a), find...

Q: Evaluate the integral by changing to cylindrical coordinates. ∫_(-3

Evaluate the integral by changing to cylindrical coordinates. ∫_(-3)^3 ∫_0^(√(9-x^2 ) ∫_0^(9-x^2-y^2) √(x^2+y^2 ) dz dy dx