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Question 1 [4]

Use Lagrange multipliers to find the point (a,b) on the graph of y = e6x, where the value of ab is as small as possible.
Question 2 [4]

Let f(x,y) = x2ex2 and let R be the triangle bounded by the lines x = 4, x = y/2, and y = x in the xy-plane. Express RR f dA as a double integral in two different ways (dxdy and dy dx), then evaluate one of your integrals to find the value of RR f dA.
Question 3 [2]

Convert the integral

to polar coordinates.
Question 4 [4]

Find the volume of the solid in R3 bounded by y = x2, x = y2, z = x + y + 24 and z = 0.
(Show calculations)
Question 5 [3]

Set up the integral RRRW f(x,y,z)dV for the function f(x,y,z) = z and region x2 + y2 = z = 49 in cylindrical coordinates.
Question 6 [3]

Convert the integral

to spherical coordinates.

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