By William Ma — McGraw-Hill Professional

Updated on Oct 24, 2011

Review the following concepts if needed:

- The Function for the Second Fundamental Theorem of Calculus for AP Calculus
- Approximating the Area Under a Curve for AP Calculus
- Area Under a Curve for Definite Integrals for AP Calculus
- Area Between Two Curves for Definite Integrals for AP Calculus
- Find the Volume for Solids with Known Cross Sections for AP Calculus
- The Disc Method for Volumes of Solids for AP Calculus
- The Washer Method for Volumes of Solids for AP Calculus
- Integration of Parametric, Polar, and Vector Curves for AP Calculus

### Part A–The use of a calculator is not allowed.

- Let where the graph of
*f*is given in Figure 12.7-1. - Evaluate
*F*(0),*F*(3), and*F*(5). - On what interval(s) is
*F*decreasing? - At what value of
*t*does*F*have a maximum value? - On what interval is
*F*concave up? - Find the area of the region(s) enclosed by the curve
*f (x)*=*x*^{3}, the*x*-axis, and the lines*x*= – 1 and*x*=2. - Find the area of the region(s) enclosed by the curve
*y*= |2*x*– 6|, the*x*-axis, and the lines*x*= 0 and*x*= 4. - Find the approximate area under the curve from
*x*= 1 to*x*= 5, using four right-endpoint rectangles of equal lengths. - Find the approximate area under the curve
*y = x*+ 1 from^{2}*x*= 0 to*x*= 3, using the Trapezoidal Rule with*n*= 3. - Find the area of the region bounded by the graphs
- Find the area of the region bounded by the curves
*x = y*^{2}and*x*= 4. - Find the area of the region bounded by the graphs of all four equations:
- Find the volume of the solid obtained by revolving about the
*x*-axis, the region bounded by the graphs of*y*=*x*^{2}+ 4, the*x*-axis, the*y*-axis, and the lines*x*= 3. - The area under the curve from
*x*= 1 to*x*=*k*is 1. Find the value of*k*. - Find the volume of the solid obtained by revolving about the
*y*-axis the region bounded by*x*=*y*^{2}+ 1,*x*= 0,*y*= – 1, and*y*= 1. - Let
*R*be the region enclosed by the graph*y*= 3*x*, the*x*-axis and the line*x*= 4. The line*x*=*a*divides region*R*into two regions such that when the regions are revolved about the*x*-axis, the resulting solids have equal volume. Find*a*. - Find the volume of the solid obtained by revolving about the
*x*-axis the region bounded by the graphs of*f (x) = x*and^{3}*g (x) = x*^{2}. - The base of a solid is a region bounded by the circle
*x*^{2}+*y*^{2}= 4. The cross of the solid sections are perpendicular to the*x*-axis and are equilateral triangles. Find the volume of the solid. - Find the volume of the solid obtained by revolving about the
*y*-axis, the region bounded by the curves*x*=*y*^{2}and*y*=*x*– 2. *R*_{1}about the*x*-axis.*R*_{2}about the*y*-axis.*R*_{1}about the line .*R*_{2}about the line- The function
*f (x)*is continuous on [0, 12] and the selected values of*f (x)*are shown in the table. - Find the area bounded by the curve defined by
*x*= 2 cos*t*and*y*= 3 sin*t*from*t*= 0 to*t*= π. - Find the length of the arc of
*r*= sin^{2}from θ = 0 to θ = π. - Find the area of the surface formed when the curve defined by
*x = e*sin^{t}*t*and*y*=*e*cos^{t}*t*from*t*= 0 to is revolved about the*x*-axis. - Find the area bounded by
*r*= 2 + 2 sin θ. - The acceleration vector for an object is . Find the position of the object at
*t*= 1 if the initial velocity is and the initial position of the object is at the origin. - in terms of
*k*. - A man wishes to pull a log over a 9 foot high garden wall as shown. See Figure 12.8-1. He is pulling at a rate of 2 ft/sec. At what rate is the angle between the rope and the ground changing when there are 15 feet of rope between the top of the wall and the log?
- (Calculator) Find a point on the parabola that is closest to the point (4, 1).
- The velocity function of a particle moving along the
*x*-axis is*v(t) = t cos(t*+ 1) for^{2}*t*≥ 0.- If at
*t*= 0, the particle is at the origin, find the position of the particle at*t*= 2. - Is the particle moving to the right or left at
*t*= 2? - Find the acceleration of the particle at
*t*=2 and determine if the velocity of the particle is increasing or decreasing. Explain why.

- If at
- (Calculator) given
*f (x) = xe*and^{x}*g (x)*= cos*x*, find:- the area of the region in the first quadrant bounded by the graphs
*f (x), g (x)*, and*x*= 0. - The volume obtained by revolving the region in part (a) about the
*x*-axis.

- the area of the region in the first quadrant bounded by the graphs
- Find the slope of the tangent line to the curve defined by
*r*= 5 cos 2θ at the point where

### Part B–Calculators are allowed.

For Problems 16 through 19, find the volume of the solid obtained by revolving the region as described below. See Figure 12.7-2.

Find the approximate area under the curve of *f* from 0 to 12 using three midpoint rectangles.

### (Calculator) indicates that calculators are permitted.

Solutions for these practice problems can be found at: Solutions to Areas and Volumes Practice Problems for AP Calculus

From 5 Steps to a 5 AP Calculus AB and BC. Copyright © 2010 by The McGraw-Hill Companies. All Rights Reserved.

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