Calculus Final Exam June 12, 2019 1. (8 points) Use Lagrange multipliers to find the extreme values of the the function
f(x, y, z)= xy2zsubject to the constraint x2+ y2+ z2= 4.
2. (a) (8 points) Use polar coordinates to combine the sum Z 1
1/
√ 2
Z x
√ 1−x2
xy dydx+ Z
√ 2
1
Z x
0
xy dydx+ Z 2
√ 2
Z
√ 4−x2
0
xy dydx into one double integral. Then evaluate the double integral.
(b) (8 points) Use polar coordinates to evaluate Z ∞
−∞
e−x2dx.
(c) (8 points) Evaluate "
Ω(x+ y)2dxdy whereΩ is the parallelogram bounded by the lines
x+ y = 0, x + y = 1, 2x − y = 0, 2x − y = 3.
3. (a) (8 points) Rewrite the integral Z 1
0
Z 1−x2 0
Z 1−x 0
f(x, y, z) dydzdx as an equivalent iterated integral in the five other orders.
(b) (8 points) Use the spherical coordinates (ρ, φ, θ) to find the volume of the solid S that is bounded above by the cone z2= x2+ y2, below by the xy−plane, and on the sides by the hemisphere z = q
4 − x2− y2.
4. Let F be a vector field defined by
F(x, y, z)= hsiny, xcosy + cosz, −ysinzi and let C be a parametrized curve given by
C: r(t)= hsint, t, 2ti for 0 ≤ t ≤ π/2.
(a) (8 points) Find a function f such that F= ∇ f.
(b) (4 points) Evaluate Z
C
F · dr.
Continued on the back
Calculus Final Exam June 12, 2019 5. (a) (8 points) Use the Green’s theorem to evaluate
C
(1+ 10xy + y2) dx+ (6xy + 5x2) dy
where C is the square with vertices (0, 0), (a, 0), (a, a), (0, a) oriented in the counterclockwise direction.
(b) (8 points) Let C be a simple closed plane curve that does not pass through the origin (0, 0). Show
that
C
− y
x2+ y2dx+ x
x2+ y2dy=
0 if C does not enclose the origin 2π if C does enclose the origin
6. (8 points) Evaluate "
S
F · n dS where F(x, y, z)= hz, y, xi and S is the surface given by
S : r(u, v)= hucosv, usinv, vi 0 ≤ u ≤ 1, 0 ≤ v ≤ π with upward normal n, i.e. the inner product n · (0, 0, 1) ≥ 0.
7. (a) (8 points) Use the Stokes’s theorem to evaluate Z
C
F · dr
where F(x, y, z)= h−y2, x, z2i and C is the curve of intersection of the plane y+ z = 2 and the cylinder x2+ y2= 1. (Orient C to be counterclockwise when viewed from above.)
(b) (8 points) Evaluate
Z
C
(y+ sin x)dx + (z2+ cosy)dy + x3dz where C is the parametrized curve given by
r(t)= hsint, cost, sin2ti ∀0 ≤ t ≤ 2π.
[Hint: Observe that C lies on the surface z= 2xy.]
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