M208H Exam 3 Practice problems

A. (15 pts.) Find the arclength of the path

c(t) = ([(t2)/2],[(t3)/3])

from t = 1 to t = 3

(Hint: once you have your differential of arc length, factor it.).

B. (20 pts.) Find the curl of the vector field

F(x,y,z) = (2xyz, x2-xy2, 2xyz-yz2) =(F1,F2,F3)

What does this tell us about whether or not F is a gradient vector field?

C. (20 pts.) Find the line integral of the vector field

F(x,y) = (x2-y2,y3-2xy)

along the path c(t) = (t,1-t) , 0 £ t £ 1.

D. (20 pts.) Find the area of the region D in the plane whose boundary is the parametrized curve

c(t) = (4t-t3,2t-t2), 0 £ t £ 2.

E. Find the velocity and acceleration of the parametric curve

(x(t),y(t)) = (t - sint, 1 + 2cost) .

F. Find the volume of the region T lying between the sphere r = 3 (in spherical coordinates) and the cone f = p/6.

G. (20 pts.) Find the integral of the function

f(x,y) = x2yz

over the region lying under the graph of the function z = x2 and over the region in the x-y plane with x2+y2 £ 4 and y ³ 0 .

(Hint: this is probably most easily done dz dy dx).

H. (20 pts.) Find the volume of the region lying under the graph of the function

f(x,y) = cos(x2+y2)+1

which lies over the circle of radius 3 in the x-y plane centered at the origin.

J. (20 pts.) A particle moves along a curve C in 3-space, starting at time t = 0 at the point (1,0,1), and at every time t, it's velocity vector is given by

[r\vec] ¢(t) = (2t,1,4t3)

What is the particle's position at time t = 2 ?

(Hint: how do you determine f(t), knowing f¢(t) and f(0) ?)

K. (20 pts.) Show that the vector field

[F\vec](x,y) = (2xy,x2-y2)

is a conservative vector field, find a potential function for [F\vec], and use this function to compute the line integral of [F\vec] over the parametrized curve

[r\vec](t) = (t2cost,tsin2 t) , 0 £ t £ p

L. (20 pts.) Use Green's theorem to compute the line integral of the vector field

[F\vec](x,y) = (2xy,y2-x2)

over the curve which follows the line segments from (0,0) to (2,0) to (0,1) to (0,0).

M. (20 pts.) Find the flux integral of the vector field

[F\vec](x,y,z) = (1,x,yz)

over that part of the graph of the function

z = f(x,y) = xy

which lies over the triangle in the plane with vertices (0,0),(1,0), and (1,2) (and using the upward pointing normal for the surface).

N. (25 pts.) Use a change of variables to find the integral of the function

f(x,y) = x2+x+3y

over the parallelogram P with vertices (0,0), (1,1), (3,1), and (4,2).

(Hint: Find the (linear!) map f which takes the unit square [0,1]×[0,1] to P.)

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