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

### Review Problems and Answers

- If and the graph of
*g*is shown in Figure 12.6-1. Find*f*(3). - The function
*f*is continous on [1, 5] and*f*> 0 and selected values of*f*are given below. - Set up an integral to find the area of the regions bounded by the graphs of
*y =x*and^{3}*y = x*. Do not evaluate the integral. - The base of a solid is the region bounded by the lines
*y = x, x*= 1, and the*x*-axis. The cross sections are squares perpendicular to the*x*-axis. Set up an integral to find the volume of the solid. Do not evaluate the integral. - Set up an integral to find the volume of a solid generated by revolving about the
*x*-axis the region bounded by the graph of*y = sin x*, where 0 ≤*x*≤ π and the*x*-axis. Do not evaluate the integral. - The area under the curve of from
*x = a*to*x*= 5 is approximately 0.916 where 1 ≤*a*< 5. Using your calculator, find*a*. - Find the length of the arc defined by
*x = t*^{2}and*y =3t*– 1 from^{2}*t*= 2 to*t*= 5. - Find the area bounded by the
*r*= 3+ cos θ. - Find the area of the surface formed when the curve defined by
*x*= sin θ,*y*= 3 sin θ on the interval is revolved about the*x*-axis.

Using 2 midpoint rectangles, approximate the area under the curve of *f* for *x* = 1 to *x* = 5.

*Answer:* Midpoints are *x* = 2 and *x* = 4 and the width of each rectangle

Area ≈ Area of Rect_{1}+ Area of Rect_{2}≈ 4(2)+8(2) ≈ 24.

*Answer:* Graphs intersect at *x* = – 1 and *x* = 1. See Figure 12.6-2.

*Answer:* Area of cross section = *x*^{2}.

*Answer:*

*Answer:* To trace out the graph completely, without retracing, we need 0 ≤ θ ≤ 2π. Then

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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