Questions from General Calculus

Q: Use Lagrange multipliers to prove that the rectangle with maximum area that

Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.

Q: Use a graphing device to draw the solid enclosed by the paraboloids

Use a graphing device to draw the solid enclosed by the paraboloids z = x2 + y2 and z = 5 - x2 - y2.

Q: Evaluate the triple integral. ∭T xz dV, where

Evaluate the triple integral. ∭T xz dV, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1)

Q: Evaluate the integral by changing to cylindrical coordinates. ∫_(-2

Evaluate the integral by changing to cylindrical coordinates. ∫_(-2)^2 ∫_(-√(4-y^2 ))^(√(4-y^2 ) ∫_(√(x^2+y^2 ) ^2 xz dz dx dy

Q: When studying the spread of an epidemic, we assume that the

When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the distance between them. Cons...

Q: Write five other iterated integrals that are equal to the given iterated

Write five other iterated integrals that are equal to the given iterated integral. ∫_0^1 ∫_y^1 ∫_0^z f (x,y,z) dx dz dy

Q: Evaluate the triple integral using only geometric interpretation and symmetry.

Evaluate the triple integral using only geometric interpretation and symmetry. ∭C (4 + 5x2yz2) dV, where C is the cylindrical region x2 + y2 < 4, -2 < z < 2

Q: Evaluate the triple integral using only geometric interpretation and symmetry.

Evaluate the triple integral using only geometric interpretation and symmetry. ∭B (z3 + sin y + 3) dV, where B is the unit ball x2 + y2 + z2 < 1

Q: Find the volume of the solid by subtracting two volumes.

Find the volume of the solid by subtracting two volumes. The solid enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2 + y

Q: Use a computer algebra system to find the exact value of the

Use a computer algebra system to find the exact value of the integral ∬R x5y3exy dA, where R = [0, 1] × [0, 1]. Then use the CAS to draw the solid whose volume is given by the integral.