CALCULUS AB WORKSHEET ON 7.1 Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator. 1. _____________________________________________________________________________________ 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. 2. 3.

4.

5. _____________________________________________________________________________________ Find the area bounded by the given graphs. Show all work. Do not use your calculator. 6. Graph for Problem 6 _________________________________________________________________________________ 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. 7. TURN->>>

( ) ( )23 12 and 4 4 on 3 3f x x g x x x= + = + - £ £

( ) ( )2 1, 1f x x g x x= - = - +

( ) ( )24 , 2f x x g x x= - = -

,2xy x y ==

2 , 6x y x y= = -

( ) ( ) 2220 , 5f x x x g x x x= + - = -

( ) ( )4 , 3 4f x x g x x= = +

-4 -3 -2 -1 1 2 3 4-10

10

20

30

40

Multiple Choice. All work must be shown. 8. (Calc) A particle moves along the x-axis so that at any time , its velocity is given by . What is the acceleration of the particle at time t = 4? (A) 2.016 (B) 0.677 (C) 1.633 (D) 1.814 (E) 2.978 _______________________________________________________________________________________

9. If and if

(A) (B) (C) 0 (D) (E)

_______________________________________________________________________________________ 10. The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

(A) feet per minute (B) feet per minute (C) feet per minute

(D) feet per minute (E) feet per minute

0t ³( ) ( )3 4.1cos 0.9v t t= +- -

22dy ydx

= 1 when 1, then when 2,y x x y= - = = =

23

-13

-13

23

78

-724

-724

78

2125

x

y

CALCULUS AB WORKSHEET 2 ON 7.1 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 1 - 3. 1. 2. (See figure on the right) 3. Figure for prob. 2 _____________________________________________________________________________________ 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. 4. 5. (See figure on the right) Figure for prob. 5 _____________________________________________________________________________________ 6. (No calculator) Let R be the region in the first quadrant bounded by the (4, 2) x-axis and the graphs of and as shown in the figure on the right. (a) Find the area of R by working in x’s. (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? _____________________________________________________________________________________ 7. (Calculator) Let R be the region bounded by the graphs of

as shown in the figure

on the right. Find the area of R. TURN->>>

( ) ( )2 2 1, 3 3f x x x g x x= + + = +

3 2 23 3 ,y x x x y x= - + =

2 , 2x y x y= = +

( )2ln 1 , cosy x y x= + =

( ) ( )2 , 2xf x x g x= =

y x= 6y x= -

12 , 4 , and ,x xy y yx

= = =

R

R

Multiple Choice. All work must be shown. 8. The graph of f is shown in the figure on the right.

If , for what value of x does

have a maximum? (A) a (B) b (C) c (D) d (E) It cannot be determined from the information given. _______________________________________________________________________________________ 9. In the triangle shown on the right, if increases at a constant rate of 3 radians per minute, at what rate is x increasing in units per minute when x = 3 units?

(A) 3 (B) (C) 4 (D) 9 (E) 12

_______________________________________________________________________________________ 10. (Calc) The velocity, in ft/sec, of a particle moving along the x-axis is given by the function . What is the average velocity of the particle from time t = 0 to time t = 3? (A) 20.086 ft/sec (B) 26.447 ft/sec (C) 32.809 ft/sec (D) 40.671 ft/sec (E) 79.342 ft/sec

( ) ( )xa

g x f t dt= ò ( )g x

q

154

( ) t tv t e te= +

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. 1. Find the volume of the solid whose base is bounded by the graphs of and , with the indicated cross sections taken perpendicular to the x-axis. (a) Squares (b) Rectangles of height 1 (c) Semiellipses of height 2 (The area of an ellipse is given by the formula , 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 2. Find the volume of the solid whose base is bounded by the circle 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 3. The base of a solid is bounded by Find the volume of the solid for each of the 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 _____________________________________________________________________________________ 4.

1y x= + 2 1y x= -

A abp=

2 2 4x y+ =

3, 0, and 1.y x y x= = =

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. 1. Let R be the region bounded by the graphs of (a) Find the area of R. (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.

2. Let R be the region bounded by the graphs of .

(a) Find the area of R. (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. 3. Let R be the region bounded by the graphs of . Find the area of R. 4. Let R be the region bounded by the graphs of . For this solid, each cross section perpendicular to the x-axis is an equilateral R triangle. Find the volume of the solid. 5.

22 and .y x y x= =

and 2xy x y= =

216 and 4x y x y= - = -

[ ]sin on 0,y x p=

CALCULUS AB WORKSHEET ON 7.2 – VOLUME BY DISCS & WASHERS Work the following on notebook paper. Do not use your calculator on problems 1 – 3. 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. 1. about the x-axis 2. about the x-axis 3. about the x-axis _____________________________________________________________________________________ Use your calculator on problems 4 – 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. 4. 5. 6. (a) about the line (a) about the x-axis (a) about the x-axis (b) about the line (b) about the line (b) about the line

(c) about the line (c) about the line _____________________________________________________________________________________ Multiple Choice. Show your work.

7. is

(A) (B)

(C) (D) (E) _______________________________________________________________________________________ 8. Let f be a continuous function on the closed interval . If then the Intermediate Value Theorem guarantees that (A)

(B) for at least one c between and 6

(C) for all x between and 6 (D) for at least one c between and 6 (E) for at least one c between and 3 _______________________________________________________________________________________

9. (Calc) Let g be the function given by for . On which of the following

intervals is g decreasing? (A) (B) (C) (D) (E)

, 0, 4y x y x= = =24 and 0y x y= - =

2 3 and y x y x= =

, 0, and 4y x y x= = = 2 2 and 4y x y x x= = - 2 and y x y x= =3y =2y = - 6y = 3y =

1y = - 2y = -

( )0

ln 1limh

e hh®

+ -

( ) ( ), where lnf e f x x¢ = ( ) ( ) ln, where

xf e f x

x¢ =

( ) ( )1 , where lnf f x x¢ = ( ) ( ) ( )1 , where lnf f x x e¢ = +

( ) ( )0 , where lnf f x x¢ =

[ ]3, 6- ( ) ( )3 1 and 6 3,f f- = - =

( )0 0f =

( ) 49

f c¢ = 3-

( )1 3f x- £ £ 3-( ) 1f c = 3-( ) 0f c = 1-

( ) ( )20sinx

g x t dt= ò 1 3x- £ £

1 0x- £ £ 0 1.772x£ £ 1.253 2.171x£ £1.772 2.507x£ £ 2.802 3x£ £

CALCULUS WORKSHEET ON 7.1 – 7.2 Work the following on notebook paper. On problems 1 – 2, use your calculator, and give your answers correct to three decimal places. Each answer must have supporting work. 1. (2004 Form B) Let R be the region enclosed by the graph of , the vertical line x = 10, and the x-axis. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y = 3. (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. 2. (2003) Let R be the shaded region bounded by the graphs of and the vertical line x = 1, as shown. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y = 1. (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 3 – 4. 3. Let R be the region enclosed by the graphs of (a) The region R is revolved about the horizontal line Write, but do not evaluate, an integral expression that gives the volume of the solid formed. (b) The region R is revolved about the horizontal line Write, but do not evaluate, an integral expression that gives the volume of the solid formed. (c) The region R is revolved about the vertical line Write, but do not evaluate, an integral expression that gives the volume of the solid formed. (d) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are rectangles with height 3. Write, but do not evaluate, an integral expression that gives the volume of this solid. (e) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are squares. Write, but do not evaluate, an integral expression that gives the volume of this solid.

4. Let R be the region enclosed by the graphs of

(a) Find the area of R. (b) The region R is the base of a solid. For this solid, the cross sections R 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 Write, but do not evaluate, an integral expression that gives the volume of the solid formed. TURN->>>

1y x= -

3 and xy x y e-= =

2 and 2 .y x y x= =5.y =

1.y = -

1.x = -

2 41

, , and .y xy yx

= = =

.3x = -

Multiple Choice. Show your work. 5. If , then the graph of f has inflection points when x = (A) – 1 only (B) 2 only (C) – 1 and 0 only (D) – 1 and 2 only (E) – 1, 0, and 2 only _______________________________________________________________________________________ 6. (Calc) A particle moves along the x-axis so that any time , its acceleration is given by . If the velocity of the particle is 2 at time t = 1, then the velocity of the particle

at time t = 2 is (A) 0.462 (B) 1.609 (C) 2.555 (D) 2.886 (E) 3.346

( ) ( )( )21 2f x x x x¢¢ = + -

0t >( ) ( )ln 1 2ta t = +