Calculus Answers

Find the volume of the solid S enclosed by the cylinder x2 + y2 = 9 and the planes y + z = 5 and

 = 1. (Hint: Convert to Cylindrical/Polar Coordinates)

2.   Evaluate ∭_T  x²  dV where T is a solid tetrahedron with vertices  (0,0,0),(1,0,0),(0,1,0) and

(0,0,1).

Evaluate ∭_E  (𝑥z−y³)dV where E = { (x,y,z) ∣ -1 ≤ x ≤ 1,  0 ≤ y ≤ 2,  0 ≤ z ≤ 1 }.

A factory produces a closed rectangular parallelepiped vats having the capacity of 10 cubic meters. Find the dimensions that will make the cost of the lining a minimum?

State the Fubini’s Theorem for the Triple Integrals and list all six possible order of

integration

1. Evaluate (𝑥 − 3) 𝑑 where = , , − 1 ≤ ≤ 1, 0 ≤ ≤ 2, 0 ≤ ≤ 1 }.

Two duopolists produce and quantities of a homogenous product . The market demand of the product is given by Q= 240-2p where Q= Qa +Qb and the price of the product. The total functions of the duopolists are given by: C(Qa)= 60+4Qa and C(Qb)= 50+0.625Qb^2

a) Find the level of output that maximizes the profit of each firm and the corresponding profit

and price.

b) Find the level of output that maximizes their profit if the two firms corporate and the

corresponding profit and price.

Examine whether the second order partial derivatives of f at (0,0) exist or not if

f :R² ➡R is defined by

f(x, y) ={x²y/√x+y² , xy≠0 and 0, xy=0

Let f(x, y) ={ y³ /x² + y² if (x, y) ≠(0, 0) and 0 otherwise

Show that f is continuous but not differentiable at (0, 0).

Check whether the limit of the function

f(x, y) = 4x²y/ x^10 +3y² exist as (x, y) ➡(0, 0)