SAMPLE PROBLEMS FOR MIDTERM II PREP
MATH 0180 FALL 2015 BROWN UNIVERSITY
(1) Let Tp be the solid inside the cylinder x2 + (y − 1)2 = 1, above the plane z = 0, and below the surface
z = x2 + y 2 . Find the volume of T . Calculate the z-component z of the centroid assuming T has unit
density.
(2) Let the wire C be the portion of y = x2 given x ∈ [0, 4]. Find it’s mass m provided the density is (xy)1/3 .
(3) Find the volume V under the paraboloid z = 3(x2 + y 2 ) over the the triangle with vertices (0, 0), (1, 0),
and (1, 2) in the xy-plane.
(4) Find the area A of the region in the first quadrant of the xy-plane bounded by the curves y = 1/x, y = x,
y = 2/x, and
RR y = 2x.
(5) Evaluate R xydxdy where R is the region described by x2 + 2x + y 2 − 6y ≤ 6.
R8R2
4
(6) Evaluate 0 y1/3 4ex dxdy.
RR
(7) Evaluate R √ 2x 2 dxdy where R is the triangle with vertices (0, 0), (0, 2), and (2, 2).
x +y
p
(8) Find the surface area A of the portion of the hemisphere z = 25 − x2 − y 2 that lies above the disk
x2 + y 2 ≤ 9 in the xy-plane.
p
(9) Using spherical coordinates find the moment of inertia Iz of the solid cone x2 + y 2 ≤ z ≤ 10 around the
z-axis. Find it using cylindrical coordinates. Assume the p
density is 1.
(10) Using cylindrical coordinates find the mass m of the cone x2 + y 2 ≤ z ≤ 2 given that the density is equal
to it’s distance from the origin.
(11) Using spherical coordinates find the volume V of the solid contained within both the sphere of radius R
centered at the origin above the xy-plane (northern hemisphere) and the cone z = r.
(12) Compute the divergence and curl of the vector field F = hx2 e−z , y 3 ln(z), z cos(y)i.
(13) Find the volume V of the region outside the sphere ρ = 2 cos(φ) and inside the northern hemisphere of
radius 2 centered at the origin.
(14) Find the volume V of the curved wedge cut out from the cylinder (x − 2)2 + y 2 = 4 by the planes z = 0
and y + z = 0.
(15) Find the volume V of the parallelepiped whose base is a rectangle given by x ∈ [0, 1] and y ∈ [0, 2] in the
xy-plane, while the top lies in the plane x + y + z = 3.
(16) Find the area A of the region inside the curve r = 6 sin(θ) and outside the circle r = 3.
(17) Find the y-component y of the centroid for the radius 3 disk centered at the origin excluding the first
quadrant (with
unit density).√
R −√2 R √4−x2
R 2 R √4−x2 2
√
(18) Evaluate −2
+ −√2 x
(x + y 2 )dydx.
− 4−x2
(19) Via line integration find the moments of inertia for a wheel of radius R with constant density δ0 .
√
(20) Evaluate the line integral of x + y − z 2 along the path C going from the origin to the point (1, 1, 0) on the
2
curve y = x in the xy-plane and then from (1, 1, 0) to (1, 1, 1) on the vertical line {(1, 1, z) : 0 ≤ z ≤ 1}.
1