Calculus 2210 - Sample Final Exam:
1) (20) Verify the divergence theorem with F=xi+yj+8k and the closed surface Sis the boundary
of the region bounded above by the paraboloid z= 1 −x2−y2and bounded below by the xy-plane.
2) (20) Compute the integral
ZC
xyz dx + 2x2z dy +y6dz
where Cis the boundary of the rectangle in the plane z=ywhere −1≤x≤1 and 0 ≤y≤2. The
curve is to be traversed counter-clockwise when viewed from above.
3) (20) Compute the integral
ZC(1 + ysin z)dx + (1 + xsin z)dy + (xy cos z)dz
where Cwhich consists of the straight line from (−2,−1,π
2) to (0,1,0), followed by the curve r(t) =
tan5ti+ cos4tj+tk, 0 ≤t≤π
4.
4) (20) Compute the integral
ZC
eydx + (xsin xz)dy +zdz
where Cconsists of the straight line between the points (1,0,1) and (0,2,0).
5) (20) Compute the flux of F=xzi−2yj+ 3xkacross a sphere of radius 2.
6) (30) Find the extreme values of f(x, y) = 2x2+1
3y2−4x−5 on the region 2x2+y2≤32. Classify
any critical points you find.
7) (20) Using the chain rule, show that given any two functions f(u) and g(v) that z=f(x+at) +
g(x−at) satifies the wave equation
∂2z
∂t2−a2∂2z
∂x2= 0 .
8) (10) Find the equation of the tangent plane to x2
4+y2+z2
4= 3 at (−2,1,2).
9) (20) Find the center of mass (or centroid) of the region above the cone z=√x2+y2and below the
sphere x2+y2+z2= 4 where the density is equal to the distance from a point in the region to the
origin (0,0,0).
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