Midterm exam 3 Sample Test
(I). Calculate the integral:
R 16 R 4 x y
1
1 y + x dydx
(II).
R 2 R 2y
1. Consider the integral I = 0 0 f (x, y)dxdy, find out the integral by reversing the
order of integration(Do NOT Evaluate).
R 5 R √25−x2 p
2. Consider the integral I = 0 0
x2 + y 2 dydx, find out the integral by converting to the polar coordinates(Do NOT Evaluate).
(III). Write the equation in spherical coordinates.
1. x2 + y 2 + z 2 = 49
2. x2 − y 2 − z 2 = 1
IV. Consider the double integral I =
y = 1 − x2 and y = 4x2 − 4
1. Sketch the graph of the region B.
2. Evaluate I by letting dA = dydx
RR
B (x
− 1)dA, where B is the region bounded by
V. Consider the solid E that is bounded by z = x2 + y 2 and z = 6
1. Find a triple integral to represent the volume of the solid with dV = dzdydx (Do
NOT Evaluate).
2. Convert to cylindrical coordinates and evaluate the triple integral.
VI.Use spherical coordinates
to evaluate I =
p
z 2 = 1 and above z = x2 + y 2 .
RRR
E
2zdV , where E is region below x2 + y 2 +