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Integration

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First some review… To integrate, you must have a sum/difference or constant multiple combination of a variable to a number power (x n ). 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.

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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

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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!

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Choose one square of each type to make Tic-Tac-Toe… Comprehensio n Creative Thinking Application Creative Thinking Application Comprehensio n Application Comprehensio n Creative Thinking

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Comprehension Integrate: (Do not click the option below until EVERYONE has attempted the problem!) Answer

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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

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Comprehension Integrate: (Do not click the option below until EVERYONE has attempted the problem!) Answer

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Comprehension Answer Back to Game Board

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

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Comprehension Answer Back to Game Board

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Creative Thinking Integrate (Do not click the option below until EVERYONE has attempted the problem!) Answer

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Creative Thinking Answer Back to Game Board

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Creative Thinking Integrate (Do not click the option below until EVERYONE has attempted the problem!) Answer

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Creative Thinking Answer Back to Game Board

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Creative Thinking Integrate (Do not click the option below until EVERYONE has attempted the problem!) Answer

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Creative Thinking Answer Back to Game Board Recall…

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Application The graph of f(x) is shown: Calculate (Do not click the option below until EVERYONE has attempted the problem!) Answer

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Application Answer Recall…the area above the x-axis is positive and the area below the x-axis is negative in integration. Using geometric shapes, divide the positive region into a triangle from [-1,0], a rectangle 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 Back to Game Board

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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 x1234567 f(x)123.5200

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Application Answer Since we need 3 equal subdivisions, use: A 1 = the area from [1,3], ∆x = 2 A 2 = the area from [3,5], ∆x = 2 and A 3 = the area from [5,7], ∆x = 2. A 1 = (2)(f(2)) = 2(2) = 4 A 2 = (2)( f(4))= 2(2) = 4 A 3 = (2)( f(6)) = 2(-1) = -2 A 1 + A 2 + A 3 = 4+4+-2 = 6. Back to Game Board x1234567 f(x)123.5200

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Application The 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

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Application Answer For average temperature, “add up all the temperatures and divide by the number of temperatures accumulated.” Back to Game Board

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6.3 Definite Integrals and the Fundamental Theorem.

6.3 Definite Integrals and the Fundamental Theorem.

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