Areas and Volumes Rapid Review for AP Calculus
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 =x3 and 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 = t2 and y =3t2 – 1 from 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 Rect1 + Area of Rect2 ≈ 4(2)+8(2) ≈ 24.
Answer: Graphs intersect at x = – 1 and x = 1. See Figure 12.6-2.
Answer: Area of cross section = x2.
Answer: To trace out the graph completely, without retracing, we need 0 ≤ θ ≤ 2π. Then
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