Integration

First some review…To integrate, you must have a sum/difference

or constant multiple combination of a variable to a number power (xn).

You may need to substitute one function for another in order to manipulate the integral to be in this form!

Integration finds net area between curves by accumulating heights.

Now you’re ready to play!Get into the following groups of students –

move your desks into these groups!!E. Yoder

D. Parker

K. Campbell

M. Flannery

J. Andrejko

R. Bauters

J. Jurado

D. Silva

G. Hucks

Now you’re ready to play!Designate one person in your group to “keep

score”. This person should draw a blank Tic-Tac-Toe board on a piece of paper. Keep track of which spaces you have filled in on this board by marking an X for the ones you get correct and an O for the ones do you not.

Remember: Your goal is to make three X’s across, down, or diagonally (just like in normal Tic-Tac-Toe). First group to accomplish this, wins 10 bonus points!

Choose one square of each type to make Tic-Tac-Toe…

Comprehension

Creative Thinking

Application

Creative Thinking

Application Comprehension

Application Comprehension

Creative Thinking

Comprehension

Integrate:

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Comprehension Answer

Discuss as a class why the answer is negative.Hint: Think about the location of the graph of

cosx between π/2 and π.Back to Game Board

Comprehension

Integrate:

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Comprehension Answer

Back to Game Board

82

82

1(0) 4

161

( 2) 416

840 4096

16

Comprehension

Integrate

(Do not click an option below until EVERYONE has attempted the problem!)

Answer

Comprehension Answer

Back to Game Board

Creative Thinking

Integrate

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Creative Thinking Answer

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2

3 2

(3 2)

3 2

3 x du

u x

Creative Thinking

Integrate

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Creative Thinking Answer

Back to Game Board

Creative Thinking

Integrate

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Creative Thinking Answer

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Recall…

2 20 2

3 32 2

2 0

x xx x

2 2( 2) 2

0 3( 2) 3(2) 02 2

( 8) 8 16

ApplicationThe graph of f(x) is shown:Calculate

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Application Answer Recall…the area above the x-axis is positiveand the area below the x-axis is negative in integration.

Using geometric shapes, divide thepositive region into a triangle from [-1,0], arectangle from [0,1], and a triangle from [1,2]. Add these up! 1+2+1 = 4

Subtract the triangular region from [2,3] and the rectangular region from [3,4]. 4-.5-1 = 2.5

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Application

The function f is continuous for all 1 < x < 7.

Approximate the value of using MRAM with 3 equal subdivisions.

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

x 1 2 3 4 5 6 7f(x) 1 2 3.5 2 0 -1 0

Application Answer

Since we need 3 equal subdivisions, use:A1= the area from [1,3], ∆x = 2A2= the area from [3,5], ∆x = 2 and A3= the area from [5,7], ∆x = 2.

A1= (2)(f(2)) = 2(2) = 4A2= (2)( f(4))= 2(2) = 4A3= (2)( f(6)) = 2(-1) = -2

A1+ A2+ A3 = 4+4+-2 = 6.

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x 1 2 3 4 5 6 7f(x) 1 2 3.5 2 0 -1 0

ApplicationThe temperature T (in ˚C) recorded during a day

followed the curve where t is time and noon is t = 0. (In other words, for this day, -12 ≤ t ≤ 12.)

What was the average temperature during the day?

(Do not click the option below until EVERYONE has attempted the problem!)

Answer

Application Answer

For average temperature, “add up all the temperatures and divide by the number of temperatures accumulated.”

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