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6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)
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6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Dec 19, 2015

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Moses Dorsey

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Page 1: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

6.1 Area Between 2 CurvesWed March 18

Do Now

Find the area under each curve in the interval [0,1]

1)

2)

Page 2: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Graphing Calculator - Integrals

• This only works if you have bounds

• Math -> 9 fnint(

• Fnint(equation,x,lower,upper)

Page 3: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Test Review

• Retakes by?

Page 4: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area Between 2 Curves

• If f(x) > g(x) on the interval [a,b], then the area between f(x) and g(x) on the interval [a,b] is

Page 5: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area Between 2 Curves

• 1) Graph both functions

• 2) Decide which function is f(x) and which is g(x)

• 3) Evaluate each integral on the given interval

• 4) Subtract the 2 values

Page 6: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Examples

Find the area between the curves on the interval [0,2]

Page 7: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Example 2

• Find the area between the curves on the interval [1,3]

Page 8: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area between 2 Curves that Cross

• If the curves cross inside a given interval, we need to split it up at the intersection point.

• Set the 2 functions equal to each other to find the intersection point.– Can use the graphing calculator to find

intersection points

Page 9: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Ex

• Ex: Find the area bounded by the graphs on the interval [0,2]

Page 10: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

You try

• 1) Find the area of the region in the interval [1, 3] between the functions

• 2) Find the area in the interval [-2, 5] between the functions

Page 11: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• Hand in: Find the area between the curves on the interval [0,2]

• HW: p.361-2 #1,2, 5, 8, 9, 16

Page 12: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

6.1 Area Bounded by CurvesThurs March 19

• Do Now

• Find the area between the 2 curves on the [0,5]

• 1)

• 2)

Page 13: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: p.361 #1,2, 5, 8, 9, 16

• 1) 102

• 2) 34/3 = 11.333

• 5)

• 8) 262/3 = 87.333

• 9)

• 17) 2 – pi/2 = .429

Page 14: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area Between 2 Curves that Intersect (no given interval)

• 1) Set f(x) = g(x)

• 2) Solve for x– The two x values are your lower and upper

bounds

• 3) Evaluate the area between the 2 curves

• Calculator: use graphs to find intersection points

Page 15: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Using a Calculator

• Graph both functions

• 2nd -> calc -> intersect

• Pick the 2 curves you want to find the intersection of

• Guess: pick a point near the intersection point

Page 16: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Examples

• Find the area bounded by the graphs of

Page 17: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Ex 2

• Find the area bounded by the graphs of

Page 18: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Areas determined by 3 Curves

• If the upper or lower curve changes from one function to another, we split it up into 2 or more areas

• Ex:

Page 19: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

You try

• Find the area of the region that is enclosed between the curves

Page 20: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

You try

• Find the area bounded by the graphs of

Page 21: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• Hand in: Find the area of the region that is enclosed between the curves

• HW: p.361 #3 4 10 13 27 29 31

Page 22: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

6.1 Area PracticeFri March 20

• Do Now

• Calculate the area determined by the intersections of the curves

Page 23: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: p. 361 #3 4 10 13 27 29 31

• 3) 32/3 = 10.667

• 4) 128

• 10) 12ln6 – 10 = 11.501

• 13) 160/3 = 53.333

• 27) 64/3 = 21.333

• 29) 2

• 31) 128/3 = 42.667

Page 24: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Practice

• (green book) worksheet p.409 #7-12, 15-20, 30-34

Page 25: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• Find the area enclosed by the following

• HW: worksheet p.409 #7-12 15-20 30-34

Page 26: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

6.1 Areas with Respect to YMon March 23

• Do Now

• Find the area bounded by the following functions

Page 27: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: p.409 #7-12

• 7) 4.86

• 8) 20.65

• 9) 3

• 10) 31/6 = 5.167

• 11) 29/2 = 14.5

• 12) 12

Page 28: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: p.409 #15-20

• 15) 1/6 = .167

• 16) 36

• 17) 27/4 = 6.75

• 18) 27/4 = 6.75

• 19) 1/12 = .083

• 20) 1/3 = .333

Page 29: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: p.409 #30-34

• 30) 4/3

• 31) 1

• 32) 4

• 33) 8

• 34) 32/3

Page 30: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Integrating with Respect to Y

• If f(y) is to the right of g(y), then the area between two curves is

• 1) Set all functions x = f(y) and x = g(y)• 2) The curve on the right is f(y)• 3) Evaluate the areas and subtract

Page 31: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Revisiting Previous Example

• Let’s integrate with respect to y instead of splitting the area up into 2 areas.

Page 32: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Example 1.6

• Find the area bounded by the graphs of

Page 33: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• Hand in: Sketch and find the area bounded by the given curves. Choose the variable of integration so that the area is written as a single integral

• HW: p.363 #19-26 skip 24• Quiz Fri

Page 34: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

6.1 Area ReviewTues March 24

• Do Now

• Sketch and find the area of the region bounded by the given curves.

Page 35: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: p.363 #19-26

• 19) 1331/6 = 221.833

• 20) 64/3 = 21.333

• 21) 256

• 22) 81/2 = 40.5

• 23) 32/3

• 25) 64/4

• 26) 3

Page 36: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area Review

• Area between 2 curves• F(x) - G(x)• Finding intersection points = bounds

• Area inside 3 curves– Splitting up into 2 areas

• What are the bounds of each area?

– Integrating with respect to Y• Using Y - bounds, and Y - functions

Page 37: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Practice

• (blue book) Worksheet p.448 #1-4, 7-13 odds

Page 38: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• Journal Entry: When trying to find the area between two curves, when should we integrate with respect to x? to y? Would you rather switch between x and y, or have to split your area problem into several problems? Why?

• HW: Finish worksheet p.448 #1-6 all, 7-13 odds

• Quiz Fri March 27

Page 39: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

6.1 Quiz Review Wed March 25

• Do Now

• Find the area of the region R between the curves

• 1) y = 2x + 1, y = 0, y = -x

• 2) y = x^2, y = -x + 2, y = 0

Page 40: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: worksheet p.448 #1-6, 7-13 odds

• 1) 9/2 11) sqrt 2• 2) 22/3 13) 1/2• 3) 1 19) 37/12• 4) 10/3 21) ~5.66• 5) 32/3• 6) 9• 7) 49/192• 9) 1/2

Page 41: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area Review

• Area between 2 curves• F(x) - G(x)• Finding intersection points = bounds

• Area inside 3 curves– Splitting up into 2 areas

• What are the bounds of each area?

– Integrating with respect to Y• Using Y - bounds, and Y - functions

Page 42: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Quiz

• All types of areas

• Can use graphing calculator, but must set up the integral correctly first

• Show all work used to set up the integral

Page 43: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Practice

• (oj) worksheet p.395 #1-6 9 10 13-17

Page 44: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• How can you find the area between 2 or more curves? How can you find the bounds?

• HW: Finish worksheet p.395 #1-10 13-17

• 6.1 Quiz Fri

Page 45: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area ReviewThurs March 26

• Do Now

• Find the area bounded by the following:

• Y = 8x – 10

• Y = x^2 - 4x +10

Page 46: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area

• 1) Graph all the functions• 2) Shade the region bounded by the

functions• 3) If possible, split up into several areas• 4) For each area: Integrate higher – lower

– Bounds = intersection points (smallest – largest)

– Calculator

Page 47: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

HW Review: worksheet p.395 #1-10 13-17

• 1) 1.571 10) 0.833

• 2) 4.189 13) 16

• 3) .0833 14) 8.167

• 4) 1.333 15) 10.667

• 5) 8.533 16) 10.667

• 6) 1.467 17) 4

• 9) 0.833

Page 48: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Area Review 6 problems

• Area between 2 curves• Higher curve – lower curve• Finding intersection points = bounds

• Area inside 3 curves– Splitting up into 2 areas

• What are the bounds of each area?

– Integrating with respect to Y• Using Y - bounds, and Y - functions

Page 49: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

You try

• The area bounded by y = sqrt x, y = -sqrt x, and y = 1 - x

Page 50: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Quiz

• All types of areas

• Can use graphing calculator, but must set up the integral correctly first

• Show all work used to set up the integral

Page 51: 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

Closure

• How can you find the area between 2 or more curves? How can you find the bounds? When would you integrate with respect to x or y?

• 6.1 Quiz tomorrow


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