AP Calculus AB
Name ________________________________
UNIT 11 EXAM – Chapter 7 (Sections 7.1-7.3) REVIEW
AREA BETWEEN CURVES & VOLUME (Disk, Washer, Shell, Cross Section Methods)
Multiple Choice. You may use a Calculator.
1. Identify the definite integral(s) that represents the area of the region bounded by the graphs of y  4  x 2 and
y  x  2.
2
a.
b.
d.
2
 ( x  x  6)dx
c.
3
3
2
2
2
2
 ( x  x  6)dx
 (4  x
2
 x  2)dx
3
2
 ( x  2)dx   (4  x )dx
2
3
e. None of these
3
2. Which of the following integrals represents the volume of the solid formed by revolving the region bounded
by the graphs of y  x 3 , y  8 , and x  1 about the line x  2 ?
8
a. 2  (2  y )(1  y ) dy
3
1
2
b. 2  [8  ( x ) ]dx
2
3 2
1
8
c. 2  ( 3 y  1)(8  y)dy
1
2
d. 2  (2  x)(8  x 3 )dx
e. None of these
1
3. The base of a solid is the region in the first quadrant bounded by the line x  2 y  6 and the coordinate
axes. What is the volume of the solid if every cross section perpendicular to the y-axis is a square.
a. 15.75
b. 36
c. 18
d. 72
e. None of these
4. Find the area of the region bounded by the graphs of the equations f ( x)  sin(2 x) and g ( x)  cos( x) over
  
the interval   ,  .
 2 6
a.
33/2
8
b.
33/2
2
c.
93/2
2
d.
93/2
8
e. None of these
5. Consider f ( x)   x3  3x 2  x and g ( x)  4  x 2 .


a. Sketch a graph of the region(s) bounded by the
two curves.






b. Determine the points of intersection.






c. Set up the integrals needed to find the area of the regions bounded by the two curves. Use the
graphing capabilities to find the area bounded by these curves.
Free Response: A Graphing Calculator may be used.
6. Consider the regions bounded by the curves y  x 2  4 and y  4  e x 1 .
a. Use a graphing utility to graph the curves
y  x 2  4 and y  4  e x 1 .
4
3
2
1
b. Use a graphing utility to find the coordinates of
the points of intersection.
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-3
-4
-5
c. Write the integral and use the integration capabilities of a graphing utility to approximate the
area of the region. Round your answer to three decimal places.
d. If this region were the base of a solid in which cross sections taken perpendicular to the x-axis
were always an equilateral triangle, set up the integral that would be used to find its volume.
s2 3
Do not evaluate. (Hint: The area of an equilateral triangle with sides s is A 
)
4
Multiple Choice: No Calculator May Be Used.
7. Identify the definite integral that computes the volume of the solid generated by revolving the region
bounded by the graph of y  x 3 , and the line y  x , between x  0 and x  1 about the line x  4 .
1


1
a.   y 3  y 2 dy
2
0
1


2
b.   y 3  y dy
1
0


d.   4  y) 2 (4  y 3 ) 2 dx
1
1


c. 2  4  x 2 )(4  x6 dx
0
e. None of these
0
Free Response. No Calculator May Be Used.
8. Consider f ( y)  4 y  y 2 and g ( y )  2 y  3 graphed below.
a. Find the points of intersection of the graphs of f and g. (2 points)
b. Set up the integral that would be used to calculate the area of the region. (3 points)
c. Use the shell method to set up the integral that would be used to calculate the volume of the
solid produced by revolving the region about the line y = -4.
d. Use the disc method to set up the integral that would be used to calculate the volume of the
solid produced by revolving the region about the x = -7.
9. Consider the region bounded by f ( x)  3sin 2 x , y  0 , from x  0 to x 
a. Graph the curve.

2
.


b. Set up an integral that can be used to find the
volume of the solid created by revolving the
region about the line x  2 .






c. Set up an integral that can be used to find the volume of the solid created by revolving the
region about the line y  4 .
d. If this region were the base of a solid in which cross sections taken perpendicular to the x-axis
were always semi-circles, set up the integral that would be used to find its volume.