Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.

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1 CALCULUS WORKSHEET ON 7. Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.. f x x and g x x on x Find the area bounded by the given graphs. Draw and label a figure for each problem, and show all work. Do not use your calculator.. f x x, g x x. f x x, g x x x. y x, y 5. x y, x 6 y Find the area bounded by the given graphs. Show all work. Do not use your calculator. 6. f x x x, g x x 5 x Graph for Problem 6

2 Answers to Worksheet on 7.. x x dx x x dx... 5 x x dx... 6 x x dx y y dy... 6 x x x 5 x dx

3 CALCULUS AB WORKSHEET ON 7. Work the following on notebook paper. Find the area bounded by the given curves. Draw and label a figure for each problem, and show all work. Do not use your calculator on problems -.. f x x x, g x x. y x x x, y x (See figure on the right). x y, x y Figure for prob. Find the area bounded by the given curves. Draw and label a figure for each problem, set up the integral(s) needed, and then evaluate on your calculator. f x x, g x x. y, x (See figure on the right) 5. f x x g x 6. y ln x, y cos x Figure for prob. 5 x 7. (No calculator) Let R be the region in the first quadrant bounded by the (, ) x-axis and the graphs of y x and y 6 xas shown in the figure on the right. (a) Find the area of R by working in x s. R (b) Find the area of R by working in y s. (c) When you found your answers to (a) and (b), was it less work to work in terms of x or in terms of y? 8. (Calculator) Let R be the region bounded by the graphs of x x y, y, and y, as shown in the figure R x on the right. Find the area of R.

4 Answers to Worksheet on x x x dx x x x x dx x x x x dx y y dy.... x x at x.7... Let a a x x dx.6 x x x x x a at ,, and. Let x x a 6. a x x dx x dx x dx.6 ln x cos x at x and x Let a a cos ln (a) Area = x dx 6 xdx... (b) Area = 6 y y dy... (c) Working in terms of y was less work. x 8. x at x, and at x.6... Let a.6... x x a x x x dx dx.6 x

5 CALCULUS AB WORKSHEET ON VOLUME BY CROSS SECTIONS Work the following problems on notebook paper. For each problem, draw a figure, set up an integral, and then evaluate on your calculator. Give decimal answers correct to three decimal places.. Find the volume of the solid whose base is bounded by the graphs of y x and y x, with the indicated cross sections taken perpendicular to the x-axis. (a) Squares (b) Rectangles of height (c) Semiellipses of height (The area of an ellipse is given by the formula A ab, where a and b are the distances from the center to the ellipse to the endpoints of the axes of the ellipse.) (d) Equilateral triangles. Find the volume of the solid whose base is bounded by the circle x y with the indicated cross sections taken perpendicular to the x-axis. (a) Squares (b) Equilateral triangles (c) Semicircles (d) Isosceles right triangles with the hypotenuse as the base of the solid. The base of a solid is bounded by following cross sections taken perpendicular to the y-axis. (a) Squares (b) Semicircles (c) Equilateral triangles (d) Semiellipses whose heights are twice the lengths of their bases y x, y, and x. Find the volume of the solid for each of the

6 Answers to Worksheet on Volume by Cross Sections. (a) V x x dx 8. (b) V x x dx.5 9 V x x dx 7.69 or 8 V.59 or x x dx (c) (d). (a) 8 V x dx.667 or (b) V x dx 8.75 or 6 V x dx or V x dx.667 or (c) (d). (a) V y dy. V y dy.9 or (b) 8 8 V y dy. or (c) V y dy.57 or (d)

7 CALCULUS AB WORKSHEET ON AREA & VOLUME BY CROSS SECTIONS Work the following on notebook paper. Do not use your calculator. Draw a figure for each problem, and show your supporting work.. Let R be the region bounded by the graphs of y x and y x. (b) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a semicircle. Write, but do not evaluate, an integral expression that gives the volume of the solid. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the y-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid. x. Let R be the region bounded by the graphs of y x and y. (b) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. Write, but do not evaluate, an integral expression that gives the volume of the solid. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the y-axis is a rectangle whose height is three times the length of the base. Write, but do not evaluate, an integral expression that gives the volume of the solid.. Let R be the region bounded by the graphs of x 6 y and x y. Find the area of R.. Let R be the region bounded by the graphs of y sin x on,. For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. Find the volume of the solid. R

8 Answers to Worksheet on Area and Volume by Cross Sections. x x at x and x (a) A = (b) x x dx... V x x dx 8 y (c) V y dy x. x at x and x x (a) A = x dx... x (b) V x dx (c) V y y dy. 6 y y at y and y A = 6 y y dy V = sin... x dx

9 CALCULUS AB WORKSHEET ON 7. VOLUME BY DISCS & WASHERS Work the following on notebook paper. Do not use your calculator on problems. Draw a figure and shade the enclosed region. Then find the volume of the solid generated by revolving the given region about the given axis.. y x, y, x about the x-axis. y x y y,, and the axis about the x-axis. y x and y x about the x-axis Use your calculator on problems 6. Draw a figure and shade the enclosed region. Then find the volume of the solid generated by revolving the given region about the given axis by setting up an integral expression and then using your calculator. Give your answers correct to three decimal places.. y x, y, and x 5. y x and y x x 6. y x and y x (a) about the line y (a) about the x-axis (a) about the x-axis (b) about the line y (b) about the line y 6 (b) about the line y (c) about the line y (c) about the line y

10 Answers to Worksheet on Volume by Discs and Washers. V x dx V x dx V x x dx. (a) (b) V x dx or V x dx x x x at x and x (a) 9.5 or 88 V x x x dx.5 or (b) (c) 6 V 6 x 6 x x dx 67. or V x x x dx 5.65 or 6 x x at x and x (a) V x x dx.9 (b) V x x dx 5. (c) V x x dx 5.

11 CALCULUS WORKSHEET ON Work the following on notebook paper. On problems, use your calculator, and give your answers correct to three decimal places. Each answer must have supporting work.. ( Form B) Let R be the region enclosed by the graph of y x, the vertical line x =, and the x-axis. (b) Find the volume of the solid generated when R is revolved about the horizontal line y =. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are equilateral triangles. Find the volume of this solid. x. () Let R be the shaded region bounded by the graphs of y x and y e and the vertical line x =, as shown. (b) Find the volume of the solid generated when R is revolved about the horizontal 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 rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. Do not use your calculator on problems.. Let R be the region enclosed by the graphs of y x and y x. (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Write, but do not evaluate, an integral expression that gives the volume of this solid. (c) The region R is revolved about the x-axis. Write, but do not evaluate, an integral expression that gives the volume of the solid formed.. Let R be the region enclosed by the graphs of y, y x, and y. x (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are semicircles. Write, but do not evaluate, an integral expression that gives the volume of this solid. (c) The region R is revolved about the vertical line x. Write, but do not evaluate, an integral expression that gives the volume of the solid formed. R

12 Answers to Worksheet on (a).. A x dx 8 V 5 x dx.58 or V 8 or 7.57 x dx 8 (b) (c) x x e x a at Let x (a) A x e dx. a (b) x V e x dx. or.5 a (c) x V 5 x e dx.55 a x x at x and x (a) A x x dx (b) V x x dx (c). (a) V x x dx A y dy... ln y (b) V y dy 8 y V y dy y (c)

13 CALCULUS AB WORKSHEET ON 7. VOLUME BY SHELLS Work the following on notebook paper. Draw a figure for each problem, shade the bounded region, and show all supporting work.. (No calculator) Let R be the region bounded by graphs of volume of the solid generated when R is revolved about: (a) the y-axis (b) the line x = (c) the line x y x, y, and x. Find the. (No calculator) Let R be the region bounded by graphs of y x and y x x. Find the volume of the solid generated when R is revolved about: (a) the y-axis (b) the line x = Use your calculator on problems - 5. Draw a figure for each problem, shade the bounded region, and show all supporting work. Give decimal answers correct to three decimal places.. (Calc) Let R be the region bounded by graphs of solid generated when R is revolved about: (a) the line x = (b) the line x 5 y x y x and. Find the volume of the x. (Calc) Let R be the region bounded by graphs of y x and y. Find the volume of the solid generated when R is revolved about: (a) the y-axis (b) the line x 5. (Calc) Let R be the region bounded by graphs of y x and y x. R Find the volume of the solid generated when R is revolved about the line x.

14 Answers to Worksheet on Volume by Shells 8. (a) V xx dx x dx... 6 (b) V xx dx... 7 (c) V x x dx x dx.... x x x at x and x 6 (a) V xx x dx... (b) V xx x dx.... x x at x and x (a) (b) V x x x dx 5. V x 5 x x dx 68.8 x. x at x and x x (a) V x x dx.788 (b) 9 x V x x dx x x at x and x. Let a x V x x dx 9.6 a

15 CALCULUS AB WORKSHEET ON AREA AND VOLUME Work the following on notebook paper. Do not use your calculator.. Let R be the region bounded by the graphs of y x and y x. (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Write, but do not evaluate, an integral expression for the volume of this solid. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y = 6. (d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the vertical line x.. Let R be the region in the first quadrant bounded by the graphs of y x, the horizontal line y = 6, and the y-axis, as shown in the figure on the right. (b) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the vertical line x =. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y = 7. (d) 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 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid.. Let R be the region bounded by the x-axis, the graph of y x, and the line x =. (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.. Let R be the region in the first quadrant bounded by the graphs of y x, y, the x-axis and the x vertical line x =. (a) Find the area of the region R. (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are rectangles with height five times the length of the base. Find the volume of this solid. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y =.

16 Answers to Worksheet on Area and Volume. x x at x and x (a) A x x dx (b) V x x dx (c) (d) 9. (a) A 6 x dx (b) 6 9 (c) V 7 x dx 9... V x x dx V x x x dx V x x dx 6 y (d) V dy 6 6. (a) A xdx... h 6 (b) xdx h (c) V x dx... 8 k (d) x dx 8 k 8. x at x x 7 (a) A xdx dx... x (b) V 5 x dx dx... x 8 (c) V x dx dx x

17 CALCULUS AB REVIEW WORKSHEET ON AREA AND VOLUME Work the following on notebook paper. Problems are noncalculator.. (Modified version of 9 AB ) (No calculator) Let R be the region in the first quadrant enclosed by the graphs of y x and y x. (b) The region R is the base of a solid. For this solid, at each x, the cross section perpendicular to the 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. (d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y. (e) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the vertical line x.. Let R be the region in the first quadrant bounded by the (, ) x-axis and the graphs of y x and y 6 xas shown in the figure on the right. R (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are rectangles with a height of 5. Write, but do not evaluate, an integral expression for the volume of this solid. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are semicircles. Write, but do not evaluate, an integral expression for the volume of this solid.. Let R be the region bounded by the graphs of x y x y and. (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are equilateral triangles. Write, but do not evaluate, an integral expression for the volume of this solid. You may use your calculator on problem.. Let R be the region bounded by the graphs of f x x x and g x x 5 x. R (b) A vertical line x = k divides R into two regions of equal area. Write, but do not solve, an equation that could be solved to find the value of k. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are isosceles right triangles Graph for Problem with the hypotenuse as the base of the solid. Write, but do not evaluate, an integral expression for the volume of this solid. (d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y. (e) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the vertical line x 7. TURN->>>

18 You may use your calculator on problem Let R be the region enclosed by the graphs of y x and y x. (b) Find the volume of the solid generated when R is rotated about the horizontal line y 8. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y 5. (d) Find the volume of the solid generated when R is rotated about the vertical line x. (e) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the vertical line x 6.

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