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1 WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first quadrant bounded by the x-axis, the graph of x = 4. Which of the following integrals gives the area of R? x = y +, and the line (A) 4 ( y ) + dy (B) ( y ) 4 + dy (C) 4 ( y + ) (D) ( y ) 4 + dy (E) 4 ( y + ) 4 dy dy. Which of the following gives the area of the region between the graphs of x = to x = 3. y = x and y = x from (A) (B) 9 (C) 3 (D) 3 (E) 7 Page of 6

2 WS 8.: Areas between Curves 3. (Calculator permitted) Let R be the shaded region enclosed by the graphs of y e x y = sin 3x, and the y-axis as shown at right. =, ( ) Which of the following gives the approximate area of the region R? (A).39 (B).445 (C).869 (D).4 (E).34 =. Which of the following gives the area x of the region enclosed by the graphs of f and g between x = and x =? (A) e e ln (B) ln e + e (C) e (D) e e (E) ln e x 4. Let f and g be the functions given by f ( x) = e and g( x) Page of 6

3 4 5. (Calculator permitted) Let R be the region enclosed by the graph of y ln( cos x) the lines x = and x =. The closest integer approximation of the area of R is 3 3 (A) (B) (C) (D) 3 (E) 4 WS 8.: Areas between Curves = +, the x-axis, and Short Answer. Unless stated not to, you may use your calculator to evaluate, as long as you show your work and integral set up. 6. Find the area of the shaded region. Be sure to show your equation for the height of your representative h y. rectangle h( x ) or ( ) (a) (b) Page 3 of 6

4 WS 8.: Areas between Curves 7. Sketch the region enclosed by the given curves. Decide to slice it vertically or horizontally. Draw your representative rectangle and label its height and width. Then find the area of the region, showing your integral set up. (a) y = sin x, y = e, x =, x π x = (b) y =, x y =, x = (c) x 3 y = x x, y 3 = x( lobes) (d) x = y, x+ y = (e) y = cos x, y = sin x, x =, π x = ( lobes) (f) y = x, y = x 8. Use your calculator to identify the region enclosed by the give graphs. Find and store the points of intersection, label them on your paper, and use them in your integral set up. Then find the area of the region enclosed by the two functions. x (a) x= y, y = x (b) y = e, y = x Page 4 of 6

5 WS 8.: Areas between Curves. Find the area of the region bounded by the parabola and the x-axis. y = x, the tangent line to this parabola at x =,. Find the number b such that the line y = b divides the region bounded by the curves y = x and y = 4 into two regions with equal area. Page 5 of 6

6 Name Date Period Worksheet 8.3 Volumes Show all work. No calculator unless stated. Multiple Choice. (Calculator Permitted) The base of a solid S is the region enclosed by the graph of y = ln x, the line x = e, and the x-axis. If the cross sections of S perpendicular to the x-axis are squares, which of the following gives the best approximation of the volume of S? (A).78 (B).78 (C).78 (D) 3.7 (E) (Calculator Permitted) Let R be the region in the first quadrant bounded by the graph of y = 8 x, the x-axis, and the y-axis. Which of the following gives the best approximation of the volume of the solid generated when R is revolved about the x-axis? (A) 6.3 (B) 5. (C) 5.4 (D) 39.7 (E) / Page of

7 3. Let R be the region enclosed by the graph of y = x, the line x = 4, and the x-axis. Which of the following gives the best approximation of the volume of the solid generated when R is revolved about the y-axis. (A) 64π (B) 8π (C) 56π (D) 36 (E) 5 4. Let R be the region enclosed by the graphs of y = e x x, y = e, and x =. Which of the following gives the volume of the solid generated when R is revolved about the x-axis? x x (A) ( e e x x ) dx (B) ( e e x x ) dx (C) ( ) x x x x (D) π ( e e ) dx (E) π ( ) e e dx e e dx Page of

8 5. (Calculator Permitted) The base of a solid is the region in the first quadrant bounded by the x-axis, the graph of y = sin x, and the vertical line x =. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume? (A).7 (B).85 (C).467 (D).57 (E) Let R be the region in the first quadrant bounded by the graph of y = 3x x and the x-axis. A solid is generated when R is revolved about the vertical line x =. Set up, but do not evaluate, the definite integral that gives the volume of this solid. 3 (A) π ( x + )( 3 x x ) dx (B) π ( x+ )( 3x x ) dx (C) π ( )( 3 ) 3 3 (D) π ( 3x x ) dx (E) ( ) 3 3x x dx 3 x x x dx Page 3 of

9 Free Response 7. (Calculator Permitted) Let R be the region bounded by the graphs of y = x, (a) Find the area of R. y = e x, and the y-axis. (b) Find the volume of the solid generated when R is revolved about the line y =. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a semicircle whose diameter runs from the graph of y = x to the graph of y = e x. Find the volume of this solid. Page 4 of

10 8. (Calculator Permitted) The base of the volume of a solid is the region bounded by the curve 3π y = + sinx, the x-axis, x =, and x =. Find the volume of the solids whose cross sections perpendicular to the x-axis are the following: (a) Squares (b) Rectangles whose height is 3 times the base (c) Equilateral triangles (d) Isosceles right triangles with a leg on the base (e) Isosceles triangles with hypotenuse on the base (f) Semi-circles (g) Quarter-circles Page 5 of

11 9. (Calculator Permitted) Let R be the region bounded by the curves y = x + and y = x for x. Showing all integral set-ups, find the volume of the solid obtained by rotating the region R about the (a) x-axis (b) y-axis (c) the line x = (d) the line x = (e) the line y = (f) the line y = 3 Page 6 of

12 . (AP -4) Let R be the region in the first quadrant bounded by the graph of y = x, the horizontal line y = 6, and the y-axis, as shown in the figure below. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 7. (c) Region R is the base of a solid. For each y, where y 6, the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, and integral expression that gives the volume of this solid. Page 7 of

13 . (AP 9-4) Let R be the region in the first quadrant enclosed by the graphs of y = x and shown in the figure. y = x, as (a) Find the area of R. (b) The region R is the base of the solid. For this solid, at each x, the cross section perpendicular to the π x-axis has area A( x) = sin x. Find the volume of the solid. (c) Another solid has the same base R. For this solid, the cross sections perpendicular to the y-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid. Page 8 of

14 . (AP 8-) (Calculator Permitted) Let R be the region bounded by the graphs of y sin ( π x) 3 y = x 4x, as shown in the figure. = and (a) Find the area of R. (b) The horizontal line y = splits the region R into two parts. Write, but do not evaluate, and integral expression for the area of the part of R that is below this horizontal line. (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. (d) The region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by h( x) = 3 x. Find the volume of water in the pond. Page 9 of

15 3. (AP 7-) (Calculator Permitted) Let R be the region in the first and second quadrants bounded above by the graph of y = and below by the horizontal line y =. + x (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the x-axis. (c) The region R is the base of a solid. For this solid, the cross sections, perpendicular to the x-axis, are semicircles. Find the volume of this solid. Page of

16 x 4. (AP -) (Calculator Permitted) Let f and g be the functions given by f ( x) = e and g( x) ln (a) Find the area of the region enclosed by the graphs of f and g between x = and x =. = x. (b) Find the volume of the solid generated when the region enclosed by the graphs of f and g between x = and x = is revolved about the line y = 4. (c) Let h be the function given by h( x) = f ( x) g( x). Find the absolute minimum value of h( x ) on the closed interval x x. Show the analysis that leads to your answer., and find the absolute maximum value of ( ) h x on the closed interval Page of

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