Calculus AB
AP PROBLEMS on Area and Volume (7.1-7.2)
Use a separate sheet of GRAPH paper to do the work.
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Period____
Date_________________
2003 AB 1 and BC 1 Form B
Let f be the function given by f ( x) 4 x 2 x 3 , and let
be the line y 18 3x , where is tangent to the
graph of f. Let R be the region bounded by the graph of
f and the x-axis, and let S be the region bounded by the
graph of f, the line , and the x-axis, as shown.
(a) Show that is tangent to the graph of y f ( x)
at the point x = 3.
(b) Find the area of S.
(c) Find the volume of the solid generated when R is revolved about the x-axis.
2003 AB 1 and BC 1
Let R be the shaded region bounded by the graphs of y
x and y e 3x
and the vertical line x = 1, as shown in the figure.
(a) Find the area of R.
(b) Find the volume of the solid generated when R is revolved about the
horizontal line y = 1.
(c) The region R is the base of a solid. For this solid, each cross section
perpendicular to the x-axis is a rectangle whose height is 5 times the
length of its base in region R. Find the volume of this solid.
2002 AB 1 Form B
Let R be the region bounded by the y-axis and the graphs of y
(a)
(b)
(c)
(b)
(c)
3
1 x
2
and
y 4 2x , as shown in the figure.
Find the area of R.
Find the volume of the solid generated when R is revolved about the xaxis.
The region R is the base of a solid. For this solid, each cross section
perpendicular to the x-axis is a square. Find the volume of this solid.
2002 AB 1 and BC 1
x
Let f and g be the functions given by f (x ) e and g( x)
(a)
x
ln x .
1
and x 1.
2
Find the volume of the solid generated when the region enclosed by the graphs of f and g between
1
x
and x 1 is revolved about the line y 4 .
2
Let h be the function given by h(x ) f (x ) g( x) . Find the absolute minimum value of h(x ) on the
1
x 1, and find the absolute maximum value of h(x ) on the closed interval
closed interval
2
1
x 1. Show the analysis that leads to your answers.
2
Find the area of the region enclosed by the graphs of f and g between x
2001 AB 1
Let R and S be the regions in the first quadrant shown in the figure. The
region R is bounded by the x-axis and the graphs of y 2 x3 and
y tan x. The region S is bounded by the y-axis and the graphs of
y 2 x3 and y tan x.
(a) Find the area of R.
(b) Find the area of S.
(c) Find the volume of the solid generated when S is revolved about the
x-axis.
2000 AB 1 and BC 1
Let R be the shaded region in the first quadrant enclosed by the graphs
2
of y e x , y 1 cos x , and the y-axis, as shown in the figure.
(a) Find the area of the region R.
(b) Find the volume of the solid generated when the region R is
revolved about the
x-axis.
(c) The region R is the base of a solid. For this solid, each cross
section perpendicular to the x-axis is a square. Find the volume
of this solid.
1999 AB 2 and BC 2
The shaded region, R, is bounded by the graph of y x 2 and the line y = 4, as
shown in the figure.
(a) Find the area of R.
(b) Find the volume of the solid generated by revolving R about the x-axis.
(c) There exists a number k, k 4 , such that when R is revolved about the
line y k , the resulting solid has the same volume as the solid in part (b).
Write, but do not solve, an equation involving an integral expression that
can be used to find the value if k.
1998 AB 1
x , and the line x = 4.
Let R be the region bounded by the x-axis, the graph of y
(a) Find the area of the region R.
(b) Find the value of h such that the vertical line x = h divides the region R into two regions of equal area.
(c) Find the volume of the solid generated when R is revolved about the x-axis.
(d) The vertical line x = k divides the region R into two regions such that when these two regions are
revolved about the x-axis, they generate solids with equal volumes. Find the value of k.
1997 AB 2
Let f be the function given by f ( x) 3cos x . As shown to the right, the graph
of f crosses the y-axis at point P and the x-axis at point Q.
(a) Write an equation for the line passing through points P and Q.
(b) Write an equation for the line tangent to the graph of f at point Q. Show
the analysis that leads to your equation.
(c) Find the x-coordinate of the point on the graph of f, between points P and
Q, at which the line tangent to the graph of f is parallel to the line PQ.
(d) Let R be the region in the first quadrant bounded by the graph of f and the
line segment PQ. Write an integral expression for the volume of the solid
generated by revolving the region R about the x-axis. Do not evaluate.
1996 AB 2
Let R be the region in the first quadrant under the graph of y
(a)
(b)
(c)
1
for 4
x
x 9.
Find the area of R.
If the line x k divides the region R into two regions of equal area, what is the value of k?
Find the volume of the solid whose base is the region R and whose cross sections cut by planes
perpendicular to the x-axis are squares.
1995 AB 4 and BC 2
The shaded regions R1 and R2 shown are enclosed by the graphs
of f ( x) x 2 and g ( x) 2 x .
(a) Find the x- and y-coordinates of the three points of
intersection of the graphs of f and g.
(b) Without using absolute value, set up an expression
involving one or more integrals that gives the total area
enclosed by the graphs of f and g. Do not evaluate.
(c) Without using absolute value, set up an expression
involving one or more integrals that gives the volume of
the solid generated by revolving the region R1 about the
line y 5 . Do not evaluate.
1994 AB 2 and BC 1
Let R be the region enclosed by the graphs of y e x , y x , and the lines x = 0 and x = 4.
(a) Find the area of R.
(b) Find the volume of the solid generated when E is revolved about the x-axis.
(c) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the
solid generated when R is revolved about the y-axis.
1993 AB 3 and BC 1
Consider the curve y 2 4 x and chord AB joining points A( 4,0) and B(0, 2) on the curve.
(a) Find the x- and y-coordinates of the point on the curve where the tangent line is parallel to chord AB.
(b) Find the area of the region R enclosed by the curve and chord AB.
(c) Find the volume of the solid generated when the region R, defined in part (b), is revolved about the
x-axis.
1992 AB 5 and BC 2
Let f be the function given by f ( x) e x , and let g be the function given by g ( x) kx , where k is the
nonzero constant such that the graph of f is tangent to the graph of g.
(a) Find the x-coordinate of the point of tangency and the value of k.
(b) Let R be the region enclosed by the y-axis and the graphs of f and g. Using the results found in part (a),
determine the area of R.
(c) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the
solid generated by revolving the region R, given in part (b), about the x-axis.
1991 AB 2
Let R be the region between the graphs of y 1 sin( x) and y x 2 from x = 0 to x = 1.
(a) Find the area of R.
(b) Set up, but do not integrate an integral expression in terms of a single variable for the volume of the
solid generated when R is revolved about the x-axis.
(c) Set up, but do not integrate an integral expression in terms of a single variable for the volume of the
solid generated when R is revolved about the y-axis.
1990 AB 3
Let R be the region enclosed by the graphs of y e x , y ( x 1)2 , and the line x = 1.
(a) Find the area of R.
(b) Find the volume of the solid generated when R is revolved about the x-axis.
(c) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the
solid generated when R is revolved about the y-axis.
1989 AB 2
6 x 4 , the line y 2 x , and the yLet R be the region in the first quadrant enclosed by the graph of y
axis.
(a) Find the area of R.
(b) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the
solid generated when R is revolved about the x-axis.
(c) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the
solid generated when R is revolved about the y-axis.
SOLUTIONS:
2003 AB 1 and BC 1 Form B
2003 AB 1 and BC 1
2002 AB 1 Form B
2002 AB 1 and BC 1
2001 AB 1
2000 AB 1 and BC 1
1999 AB 2 and BC 2
1998 AB 1
1997 AB 2
1996 AB 2
1995 AB 4 and BC 2
1994 AB 2 and BC 1
1993 AB 3 and BC 1
1992 AB 5 and BC 2
1991 AB 2
1990 AB 3
1989 AB 2