AP Calc Unit 7

Unit #7: Area AP Calculus AB

AP Calculus AB Assignments

Unit #7--Area

Assignment #1: Definite Integration________________Page 278: 33-41 odd; Page 291: 5, 9, 11, 15, 17, 19, 27, 29, 31; Page 305: 71, 73, 77

(Check your integrals with your calculator.)

Assignment #2: LRAM-RRAM-MRAM________________ Handout

Assignment #3: Area Under a Curve________________ Page 278: 13-21 odd, 23-31 odd; Page 291: 33-41 odd

Assignment #4: Area Under a Curve________________Find the area bounded by the functions given.

1. y = x , y = 0, x = –3, x = 4 (25/2)2. y = –x2 + 9 , y = 0, x = –4, x = 5 (24 2/3)3. y = sin x , y = 0, x = –/2, x = (3)4. y = x2 + 2x – 4, y = 0, x = –5, x = 3 (set up only—do not evaluate)

Assignment #5: Area Between Curves________________Page 452: 1-7 odd, 17, 19, 21

Assignment #6: More Area Between Curves-dy________________Page 452: 23, 25, 27, 29, 31 (set up only), 35

Assignment #7: Absolute Value________________Handout

Assignment #8: Average Value________________Page 291: 47–50 all

Even Answers: 48: 16/3, x = ; 50: 2/, x 0.881

Assignment #9: Trapezoidal Rule________________Page 314: 1, 7, 9, 11 (Only Use Trapezoidal Rule)

Assignment #10: Review_____________

Test

Unit #7: Area AP Calculus ABTopic: Definite Integral

Goal: Evaluate definite integrals using the First Fundamental Theorem of Calculus.

Indefinite Integral:

F(x) + C

general antiderivative

Definite Integral:

There are limits of

integration: a and b. a is the lower limit and b is the upper limit.

The First Fundamental Theorem of Calculus

If f(x) is continuous on [a, b] and F(x) is the antiderivative of f(x), then

= = F(b) - F(a)

Note: The result is a number!!!!!!

There is no interpretation of the result. YET


u =

du =


u =

du =

u =

du =

u =

du =

Assignment #1: Page 278: 33-41 odd; Page 291: 5, 9, 11, 15, 17, 19, 27, 29, 31; Page 305: 71, 73, 77


Topic: Approximate Area Under a Curve

Goal: Approximate the area under a curve using LRAM, MRAM, RRAM (Reimmann Sums).

Approximating the Area Under a Curve

Rectangular Approximation Methods LRAM: Left Rectangular Approximation Method

MRAM: Midpoint Rectangular Approximation RRAM: Right Rectangular Approximation Method

ba

ba

baba


LRAM---RRAM---MRAM

Find the area of the curve bounded by y = 3x and the x-axis from x = 0 to x = 4 using n = 4 (note: this denotes the number of rectangles.)

0 4

0 4

0 4


Find the area of the curve bounded by y = x3 + 3 and the x-axis from x = –2 to x = 2 using n = 4.

LRAM

RRAM

MRAM


Use the table below to approximate the area under the curve using 3 equal intervals and LRAM, RRAM, and MRAM.

x y0 41 22 23 44 85 146 22

Assignment #2: Handout

Unit #7: Area AP Calculus ABLRAM, RRAM, MRAM—Reimman Sums

For the following functions, graph each function over the given interval. Find the approximate area bounded by each equation and the x-axis using a)LRAM, b) RRAM, and c) MRAM with four subintervals of equal length. For each, make a sketch of the rectangles associated with each method. Then, evaluate the definite integral of the function for the given interval.

1.

2.

3.

4. You and your companion are about to drive a twisty stretch of dirt road in a car whose speedometer works but whose odometer is broken. To find out how long this particular stretch of road is, you record the car’s velocity at 10-sec intervals, with the results in the table below. Estimate the length of the road using three intervals of equal length (a) using left-endpoint values, (b) using right-endpoint values, and (c) using midpoint values.

Time (sec) Velocity (converted to ft/sec)

0 010 4420 1530 3540 3050 4460 35

Page 279: 45 and 46

Answers1. 3/2, 5/2, 22. 3/4, 5/4, 13. , , 4. 900, 1900, 2460

46: (a) 64, (b) 236, (c) 136


Topic: Area (Not Negative)

Goal: To find the area under a curve using definite integrals.

The area under a curve is defined to be the integral below:

Note: Need an accurate drawing of the function and be able to identify the bounded region.

Find the area bounded by y = x2 and the x-axis (y = 0) between x = 0 and x = 1.

Set up an integral to find the area under the curve

of f(x) = , for –1 < x < 1. Set up an integral to find the area under the curve

of y = 3 – , for –3 < x < 3.

Find the area bounded by x-axis (y = 0) and y = 6 - x - x2.

Assignment #3: Hwk: Page 278: 13-21 odd, 23-31 odd and Page 291: 33-41 odd


Topic: Area (Negative)

Goal: To find the area under a curve using definite integrals.



Area bounded by y = sin x and the x-axis on the interval [0, ].

What if I wanted the area on the interval from [0, 2]?

Set up an integral to find the area bounded by the function f(x) and the x-axis shown below. Then find the area.


Find the area bounded by the graph of y = x3–4x and the x-axis.

Find the area bounded by the graph of y = x2 – 3 and the x-axis, from x = –2 to x = 4.

Assignment #4: See Assignment Sheet.


Topic: Area Between Two Curves

Goal: Find the area bounded by two curves.



Find the points of intersection of the graphs.

Find the area bounded by the graphs of y = 5, y = x2 – 4.

Area enclosed by y = 2 – x2, and y = –x.

Area bounded by y = x + 6, y = x2, x = 0, and x = 2.

Assignment #5: Page 452: 1 – 7 odd, 17, 19, 21


Topic: Area Between Two Curves

Goal: Find the area bounded by two curves.



Find the points of intersection of the graphs.

Area bounded on right by y = x – 2 on the left by x = y2 and below by the x-axis.

Find the area bounded by x = 4 – y2 and the y-axis.

Find the area bounded by x = 10 – y2 and the line x = 1.

Assignment #6: Page 452: 23, 25, 27, 29, 31, 35


Extra Practice if Needed Area Worksheet

Find the region bounded by the graphs in each problem.

1. y = x2 – 2x and y = x Answer: 9/2

2. y = x2 and y = –x2 + 4x Answer: 8/3

3. y = 7 – 2x2 and y = x2 + 4 Answer: 4

4. the x–axis, y = –x2 – 2x, between x = –3 and x = 2 Answer: 28/3

5. the x–axis, y = 3x2 – 3, between x = –2 and x = 2 Answer: 12

6. the x–axis, y = x3 – 3x2 + 2x, 0 < x < 2 Answer: 1/2

7. the x–axis, y = x3 – 4x, –2 < x < 2 Answer: 8

8. y = 1, y = cos2x, 0 < x < Answer: /2

9. x = y3 and x = y2 Answer: 1/12

10. x = 2y2 – 2y and x = 12y2 – 12y3 Answer: set up-only or (1765)/(1296)

11. y = 2x2 and y = x4 – 2x2 Answer: 128/15

12. y = x2 and y = –2x4 , –1 < x < 1 Answer: 22/15

13. on the left by y = x, below by y = x2/4, above by y = 1 Answer: 5/6

14. y = x2, x + y = 2 , and the x–axis Answer: 5/6

15. y = 2x – x2, y = –3 Answer: 32/3

16. y = 2sinx and y = sin 2x, 0 < x < Answer: 4

17. y = sec2x y = tan2x, x = –/4, and x = /4 Answer: /2


Topic: Absolute Value

Goal: Evaluate integrals with absolute values.

An accurate graph will make it a lot easier to setup and evaluate the integrals needed.

Evaluate

Assignment #7: Absolute Value Handout


Absolute Value Integration

1. 2. 3.

4. 5. 6.

7. 8. 9.

1. 5/2 2. 5/2 3. 17/2

4. 5 5. 9/2 6. 73/2

7. 65/2 8. 1 9. 44/3


Topic: Average Value

Goal: Find the average value over an interval.

If f is integrable on the closed interval [a, b], then the average value of f on the interval is

Find the average value of f(x) = 3x2 –2x on the interval [1, 4].

Find the average value of f(x) = cos x on the interval [0, /2].

Assignment #8: Page 291: 47-50 all


Topic: Trapezoidal Rule

Goal: Approximate the area under a curve using the trapezoidal rule.

Formula for the area of a

trapezoid is: A = h(b1+b2).

Use the trapezoid rule with four trapezoids to approximate the area under each curve.

x 1 4 6 10 15f(x) 5 12 15 20 22

Assignment #9: Page 314: 1, 7, 9, 11 (Only Use Trapezoidal Rule)

x y

Unit #7: Area AP Calculus ABUnit #7 Review

Find the total area of the region with the given boundaries. Include a graph for each problem and SHADE the appropriate region. Show all work—every step.

1. Find the area bounded by the graphs of y = x2 and y = .

2. Find the area bounded by the graphs of y = x, y = , and x = 2.

3. Find the area bounded by the graphs of x = y2 + y and x = y3. Set up Only

4. Find the area bounded by the graphs of x = sin y and x = cos y, y = , and y = .

5. Approximate if n = 4 using:

LRAM: RRAM: MRAM: TRAP:

Evaluate the following integrals

6. 7.

8. 9.

Graph the function. Find its average value over the given interval. At what point or points in the given interval does the function assume its average value?

10.

Unit #7: Area AP Calculus ABAnswers

Find the total area of the region with the given boundaries. Include a graph for each problem and SHADE the appropriate region. Show all work—every step.

1. Find the area bounded by the graphs of y = x2 and y = .

2. Find the area bounded by the graphs of y = x, y = , and x = 2.

3. Find the area bounded by the graphs of x = y2 + y and x = y3.


4. Find the area bounded by the graphs of x = sin y and x = cos y, y = , and y = .

5. Approximate if n = 4 using:

LRAM: 2[1 + 25 + 193 + 649] = 1736

RRAM: 2[25 + 193 + 649 + 1537] = 4808

MRAM: 2[4 + 82 + 376 + 1030] = 2984

TRAP: (1/2)2[1 + 2(25) + 2(193) + 2(649) + 1537] = 3272

x y0 12 254 1936 6498 1537

x y1 43 825 3767 1030

Unit #7: Area AP Calculus ABEvaluate the following integrals

6.

7.

8.

9.

Graph the function. Find its average value over the given interval. At what point or points in the given interval does the function assume its average value?

10.

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AP Calc Unit 7