Chapter 10 Review Practice
1. Consider the parametric curve x(t)=et and y(t)=2et , t≥0
a) Compute the length of the curve from t=0 to t=1
b) Convert this parametric equation into an equation of the form y=f(x) and compute the length
of the graph of f(x) from x-1 to x=2.
2. Consider the parametric curve x(t)=acos t and y(t) = bsin t, 0≤t≤pi for a≠0 and b≠0
a) Sketch and identify this curve.
b) Set up an integral to compute the length of this curve.
3. The curve below can be parametrized by x(t)=sin t and y(t)= t+cos t. Give parametrization
for the curves B and C below.
A
B
C
4. Match each of the rectangular graphs of r=f() in the left column with the corresponding
polar graph in the right column. Explain your reasoning.
1

,0   
cos  sin 
4
a) What happens to r as  approaches π/4
b) Convert the equation to an equation with rectangular coordinated.
c) What is the domain of this polar curve when viewed as a function y=f(x)

5. Consider the polar curve r 
6. Consider the polar curve r=f() shown below. Note that 2≤ f()≤5 for all theta and
f(0)=f(2π)
a) Show that the area enclosed by the graph of r=f() must be greater than 10
b) Must the area enclosed by this curve be finite for 0≤≤2π? Why or why not?
c) Find a function g() satisfying the same conditions as f() above such that the area
enclosed by g() is 14π
7. Consider the curve x=et-5cos t and y(t) =et-5sin t 0≤≤2π
a) Find the values of t where the line tangent to the curve is vertical.
b) Find the values of t where the slope of the line tangent to the curve is -1
8. Find the area under the parametric curve x(t)=sin t, y(t)= cos t sint, 0≤t≤π
9. Consider the parametric curve x=3t5+1, y=10t3-1 t 
a) Find the values of t for which the slope of the line tangent to the curve is 1.
b) What value of t gives the point (1,-1) on this curve?
c) Is dy/dx defined at (1,-1)? Describe the shape of the curve at this point.

10. Consider the parametric curve x=tcos t and y=tsin t , t>0
a) Write an integral that describes the length of this curve from t=0 to t=π
b) What substitutions couls be used to evaluate the integral in part a. (Do not evaluate the
integral