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AP Calculus BC Friday, 30 January 2015 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. TESTS are not graded. Get an 89 calculator and an 84 calculator. AP EXAM REGISTRATION (through 03/04/15) www.TotalRegistration.net/AP/443381

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Sec. 7.1: Area of a Region Between Two Curves Curves may give different regions. Ex:Find the bounded area between f (x) = –3x 3 – 2x 2 + 5x and g (x) = –3x (–2, 6) (0, 0) ( 4 / 3, –4) f (x) g (x)

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Sec. 7.1: Area of a Region Between Two Curves Ex:Find the area of the region bounded by x = y 2 – 5 and y = x – 1 (–1, –2) (4, 3) (–5, 0) x = y 2 – 5 y = x – 1

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Sec. 7.1: Area of a Region Between Two Curves Ex:Find the area of the region bounded by x = y 2 – 5 and y = x – 1. (–1, –2) (4, 3) (–5, 0)

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Stopped here 1 st Period on Monday, 03 February 2014. Stopped here 2 nd Period on Monday, 03 February 2014.

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Sec. 7.1: Area of a Region Between Two Curves Ex:Find the area of the region bounded by x = y 2 – 5 and y = x – 1 It is sometime advantageous to integrate with respect to y. (–1, –2) (4, 3) (–5, 0) x = y 2 – 5 y = x – 1

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Sec. 7.1: Area of a Region Between Two Curves In general – Vertical Rectangles Horizontal Rectangles

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Sec. 7.1: Area of a Region Between Two Curves The consumption (in billions of barrels) of petroleum in the United States from 1960 to 1979 can be modeled by where t = 0 corresponds to January 1, 1970. From 1979 through 1990, the consumption can be modeled by Find the total amount of fuel saved from 1979 through 1990 as a result of fuel being consumed at the post 1979 rate rather than at the pre-1979 rate.

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Sec. 7.1: Area of a Region Between Two Curves t = 9t = 20 The total amount of fuel saved from 1979 through 1990 was 21.674 billion barrels of petroleum.

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AP Calculus BC Tuesday, 31 January 2012 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. TESTS are not graded. YOU WILL NEED A TI-89 (from the table) AND A TI-84. –Sixth Period will need to install batteries.

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Sec. 7.2: Volume: The Disk Method

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Some examples of solids of revolution:

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Sec. 7.2: Volume: The Disk Method If a region in the plane is revolved about a line, the resulting solid is a solid of revolution.

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Sec. 7.2: Volume: The Disk Method When we put these disks together, we get a solid.

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Stopped here 2 nd Period on Friday, 30 January 2015.

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AP Calculus BC Monday, 02 February 2015 OBJECTIVE TSW (1) find the volume of a solid of revolution using the disk method, and (2) find the volume of a solid of revolution using the washer method. TESTS are graded. ASSIGNMENT DUE DATES –Sec. 7.1 Tuesday, 02/03/15 –Sec. 7.2: Disc Method Wednesday/Thursday, 02/04- 05/15 –Sec. 7.2: Washer Method Wednesday/Thurday, 02/04- 05/15 AP EXAM REGISTRATION (through 03/04/15) www.TotalRegistration.net/AP/443381

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Sec. 7.2: Volume: The Disk Method

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Ex:Find the volume of the solid of revolution by rotating about the x-axis to the line x = 9. x = 9 (0, 0)

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AP Calculus BC Monday, 02 February 2015 OBJECTIVE TSW (1) find the volume of a solid of revolution using the disk method, and (2) find the volume of a solid of revolution using the washer method. TESTS are graded. ASSIGNMENT DUE DATES –Sec. 7.1 Tuesday, 02/03/15 –Sec. 7.2: Disc Method Wednesday/Thursday, 02/04- 05/15 –Sec. 7.2: Washer Method Wednesday/Thurday, 02/04- 05/15 AP EXAM REGISTRATION (through 03/04/15) www.TotalRegistration.net/AP/443381

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Stopped here 1 st Period on Friday, 30 January 2015.

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Sec. 7.2: Volume: The Disk Method Ex:Find the volume of the solid of revolution by rotating the region formed by y = 3 – x 2 and y = 2 about the line y = 2. y = 2 y = 3 – x 2 (–1, 2) (1, 2)

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Ex:Find the volume of the solid of revolution by rotating the bounded region formed by f (x) = x 3, the y-axis, and y = 8 about the y-axis. Sec. 7.2: Volume: The Disk Method y = 8 f (x) = x 3 (0, 0) (2, 8)

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Jersey Village Falcons

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AP Calculus BC Friday, 08 February 2013 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. TESTS will be given back Monday.

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Stopped here 2 nd Period on Thursday, 07 February 2013.

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AP Calculus BC Thursday, 06 February 2014 OBJECTIVE TSW (1) find the volume of a solid of revolution using the washer method, and (2) find the volume of a solid with known cross sections. TESTS are not graded. You will need a TI-83/84 (mine or yours). AP EXAM REGISTRATION (through 03/07/14) www.TotalRegistration.net/AP/443381

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Sec. 7.2: Volume: The Washer Method

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Stopped here 2 nd Period on Monday, 02 February 2015.

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NOTE: To link to SketchPad on the next slide, the program AND file must be already open.

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Sec. 7.2: Volume: The Washer Method Sketchpad: Solids of Revolution

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Sec. 7.2: Volume: The Washer Method

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AP Calculus BC Monday, 06 February 2012 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. You will need a TI-89 and a TI-84.

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Sec. 7.2: Volume: The Washer Method Ex:Find the volume of the solid formed by rotating the bounded region of and y = x 2 about the x-axis.

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Stopped after the previous problem for 6 th period on Friday, 03 February 2012.

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Stopped after the previous problem for 7 th period on Wednesday, 01 February 2012.

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Sec. 7.2: Volume: The Washer Method The Washer Method If R(x) is the outer radius and r(x) is the inner radius of a solid of revolution, then the volume of the solid of revolution on [a, b] is Another way to look at this is as the Marshall Plan: named in honor of Ashley Marshall, JVHS c/o 1999.

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Showed the slide (just the start) of the next problem 1 st Period on Wednesday, 06 February 2013.

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Sec. 7.2: Volume: The Washer Method Ex:Find the volume of the solid formed by rotating the bounded region of and y = x/4 about the x-axis.

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Stopped here 1 st Period on Monday, 02 February 2015.

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Sec. 7.2: Volume: The Washer Method On a blank sheet of paper (to be turned in), for each problem, (a) draw a picture (label, shade, representative rectangle, axis of rotation), (b) set up the integral, and (c) evaluate. Use exact values, leaving π in your answer. wire basket Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. (1)the x-axis (2)the y-axis (3)the line x = 4 (4)the line x = 6

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Sec. 7.2: Volume: The Washer Method On a blank sheet of paper (to be turned in), for each problem, (a) draw a picture (label, shade, representative rectangle, axis of rotation), (b) set up the integral, and (c) evaluate. Use exact values, leaving π in your answer. wire basket Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. (1)the x-axis(2)the y-axis (3)the line x = 4(4)the line x = 6

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7.3 VOLUMES. Solids with Known Cross Sections If A(x) is the area of a cross section of a solid and A(x) is continuous on [a, b], then the volume of the.

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