Math 129
Section 8.2
1. Find the volume of the solid whose base is the region in the xy-plane bounded by the
first arch of y=sin x and y=-sin x and whose cross sections perpendicular to x –axis are
squares with one side in the xy plane.
2*. Find the volume of the solid whose base is the region in the xy-plane bounded by the
curves y= x2 and y=8-x2 whose cross sections perpendicular to x –axis are squares with
one side in the xy plane.
3. Consider the region bounded by the curve y=ex the x-axis and the lines x=-,1 and
x=1. Find the volume of the following solids:
A. The solid obtained by rotating the region about the x-axis.
B. The solid obtained by rotating the region about the horizontal line y=-5.
4*. Find the exact volume of the solid obtained by revolving the region between the
graph of y=4-x2 and the x-axis about the line y=5.
5*. The region bounded by y=x3, x=2, and y=-1 is revolved about the line ,y=-8. Sketch
the picture of the solid, and find its volume.
6*. Sketch a picture of the solid obtained by revolving the region bounded by the curve
y=x4 and the line y=x about the y-axis. Also find the volume of the solid.
7. Recall that the arc length of the function is f(x) from (a,f(a)) to(b,f(b)) is given by
b

1  ( f ' ( x)) 2 dx
a
Use this to approximate the length of the curve y=x3 from (0,0) to (2,8) accurate to one
decimal place. The approximation must include a strict lower bound and a strict upper
bound. Explain how you determined your answer.
8*. Recall that the arc length of the function is f(x) from (a,f(a)) to(b,f(b)) is given by
b

1  ( f ' ( x)) 2 dx
a
Use this to approximate the length of the curve y=lnx from (1,0) to (4,ln4) accurate to at
lest two decimal places. The approximation must include a strict lower bound and a strict
upper bound. Explain how you determined your answer.
9. Calculate the exact value of the arc length of the function f ( x)  4  x 2 between the
points where x=0 and x=2.
10. Consider the region bounded by the curve y  3 x , the x-axis, and the lines x=0,
and x=1. Find the volume of the following solids:
A. The solid obtained by rotating this region about the x-axis.
B. The solid obtained by rotating this region about the horizontal line y=-2.
C. The solid obtained by rotating this region about the vertical line x=1.
11. John starts at the origin and walks along the graph of y=f(x)=
x2
at the velocity of
2
10 units per second.
A. Write the integral which shows how far John has traveled when he reaches the
point where x=a.
B. You want to find the x-coordinate of the point John reaches after traveling for
2 seconds. Find an upper and lower estimate, differing by less than 0.2.