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The experimental verification of a theory concerning any natural phenomenon generally rests on the result of an integration… J. W. Mellor…

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 § A§ A§ A§ Area of a Region Between two Curves If f and g are continuous on [a, b] and g (x) f (x) for all x in [a, b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is (Upper Function – Lower Function)

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 § Area of a Region Between two Curves f (x) g (x)

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 § Area of a Region Between two Curves f (x) g (x) f (x) g (x)

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 E§ EE§ Example. Find the area between the two given curves From: P P P P P aaaa uuuu llll '''' ssss O O O O nnnn llll iiii nnnn eeee M M M M aaaa tttt hhhh N N N N oooo tttt eeee ssss

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AP Calculus Area of a Region Between Two Curves Larson – Hostetler – Edwards: Chapter 7.1 Practice. Textbook 7.1 pp. 452-455, prob. 1, 3, 5, 13, 17, 21, 23, 27, 33, 37, 39, 43, 45, 47 Suggested Extra Practice : pp 452-455, prob. 2, 4, 6, 14, 18, 22, 24, 28, 34, 38, 40, 44, 46, 48

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The moving power of mathematical invention is not reasoning but imagination … Augustus de Morgan…

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This is a solid obtained by rotating a region in the plane about an axis (called axis of revolution). § S§ S§ S§ Solids of Revolution AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 This is a solid obtained by rotating a region in the plane about an axis (called axis of revolution). § Solids of Revolution

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 The simplest solid of revolution is a right circular cylinder or disk, which is formed by revolving a rectangle about an adjacent axiswr wr The corresponding volume is: Volume disk = area disk width disk V = rrrr2 w r = radius of the disk w w = width

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§ Solids of Revolution AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 To find the volume of any solid of revolution, consider this as composed of n-disks of width w = x and radius r = r ( xi ) An approximation for the volume of the solid will be:

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§ Solids of Revolution AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 Increasing the number n of disks ( i. e. x 0 ), the approximation to the volume becomes better. The volume of the solid will be given by the limit process:

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§ T§ T§ T§ The Disk Method AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 If r(x) is continuous and r(x) > 0 on [a, b], then the volume obtained by rotating the region under the graph, can be calculated using one of the following formulas. Horizontal Axis of Revolution: Vertical Axis of Revolution:

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Show that the volume of a sphere of radius r is xy0 r r r r r xy

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Show that the volume of a right cone of radius r and height h is given by the formula : xy0 h r xy

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating about the x-axis the region between x = 0, x = 1, and under the curve

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating about the x-axis the region between the lines x = 1, x = 3, and under the curve

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating the region bounded by y = x 3, y = 8, and x = 0 a a a about the y-axis

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating about the y-axis the region bounded by x = 4, x = 0 and

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 Practice. Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, 9, 11, 13 Suggested Extra Practice : pp 463-466, prob. 2, 4, 6, 8, 10, 12, 14

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§ T§ T§ T§ The Washer Method AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 Method useful to find the volume of a solid of revolution with a hole, by changing the disk with a washer. The washer is formed by revolving a rectangle between to curves around the x-axis or any axis of the form x = awR r wR r V = (R2 – r2)w Volume of Washer

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§ The Washer Method AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 If R(x) and r(x) are continuous functions on [a, b], R(x) > 0, r(x) > 0, and R(x) > r(x), then the volume obtained by rotating the region between R(x) and r(x) can be calculated using one of the following formulas: Horizontal Axis of Revolution: Vertical Axis of Revolution:

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating the region between the lines f(x) = x2 + 3, and g(x) = x2 + 1, around the x- axis

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating the region between the lines f(x) and g(x), around the x- axis

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating the region between the lines f(x) and g(x), around the axis y = 1

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid obtained by rotating the region between the lines f(x), g(x) and x = 1, around the axis y = 2

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid formed by revolving the region bounded by the graph of y = x2 + 1, y = 0, x = 0 and x = 1, about the y - axis

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid formed by revolving the region bounded by the graph of y = 3/x, and y = 4 x, a a a about x = 1

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 Practice. Textbook 7.2 pp. 463-466, prob. 15, 17, 19, 21, 23, 27, 29 Suggested Extra Practice : pp 463-466, prob. 16, 18, 20, 22, 24, 28, 30

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§ S§ S§ S§ Solids with Known Cross Sections AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 With the disk method it is possible to find the volume of a solid with a circular cross section: R(x)]2 = A(x) This method can be generalized to solids with any shape, as long as a formula for the area of the arbitrary cross section is known

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§ Solids with Known Cross Sections AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 Area of Cross Section = A(x)

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§ V§ V§ V§ Volumes of Solids with Known Cross Sections AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 1. For cross sections of area A(x) taken perpendicular to the x-axis: 2. For cross section of area A(y) taken perpendicular to the y-axis

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid with semicircular cross section and whose base is bounded by the graphs of y = 0, x = 9 and From: D DEMOS with POSITIVE IMPACT, NSF DUE 9952306 r

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid with square cross section and whose base is bounded by the graphs of From: D DEMOS with POSITIVE IMPACT David R. Hill, Temple University - Lila F. Roberts, Georgia College & State University

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid with equilateral triangle cross section and base bounded by the graphs of y = sin x, x = 0, x = sin x From: D DEMOS with POSITIVE IMPACT David R. Hill, Temple University - Lila F. Roberts, Georgia College & State University

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid with square cross section and base bounded by the graphs of x = 0, x = y = 0 and From: D DEMOS with POSITIVE IMPACT David R. Hill, Temple University - Lila F. Roberts, Georgia College & State University

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 E§ EE§ Example. Find the volume of the solid with equilateral triangle cross section and base bounded by the circle of radius 2, centered at (0, 0) From: D DEMOS with POSITIVE IMPACT David R. Hill, Temple University - Lila F. Roberts, Georgia College & State University xy

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AP Calculus Volume: The Disk Method Larson – Hostetler – Edwards: Chapter 7.2 Practice. Textbook 7.2 pp. 463-466, prob. 61, 62, 63, 64 Suggested Extra Practice : pp 463-466, prob. 61, 62, 63, 64

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The advancement and perfection of mathematics are intimately connected with the prosperity of the State … Napoleon…

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This is an alternative method for finding the volume of a solid of revolution Compared with the disk method, the representative rectangular section is parallel to the axis of revolution § T§ T§ T§ The Shell Method AP Calculus Volume: The Shell Method Disk Method Larson – Hostetler – Edwards: Chapter 7.3 wr wr Shell Methodhw p hp

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§ The Shell Method AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 Disk MethodShell Method From: D DEMOS with POSITIVE IMPACT

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§ The Shell Method AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 Volume of Shell: h w pyx ph p + w / 2 p w / 2 w y x If w w = y, p = p(y), h = h(y) V = 2[ p(y) h(y)]y

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§ The Shell Method AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 The volume of the solid obtained by revolving the area under function h, can be calculated in the next form: Vertical axis of revolution Horizontal axis of revolution p(y) = distance to axis p(x) = distance to axis

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Find the volume of the solid obtained by rotating the area between the graph of f(x) = 1 – 2x – 3x2 – 2x3 and the x-axis over [0, 1] about: (a) the y-axis, (b) x = 1 Rogawski, 402 (a)(a)(a)(a) (b)(b)(b)(b)

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Calculate the volume of the solid obtained by rotating the area enclosed by the graphs of f(x) = 9 – x2 and g(x) = 9 – 3x about: (a) x-axis, (b) y = 10 Rogawski, 403 (a)(a)(a)(a) (b)(b)(b)(b)

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Calculate the volume of the solid obtained by rotating the region between the graph f (y) = 12(y2 – y3) and the y-axis, over [0, 1], about: (a) the x-axis, (b) the line y = 1 Finney, 392 (a)(a)(a)(a) (b)(b)(b)(b)

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Calculate the volume of the solid obtained by rotating the region between the graphs of f (x) = y 4/4 – y 2/2 a a a and x = y 2/2 about: (a) x-axis, (b) y = 2 Stewart 395 (a)(a)(a)(a) (b)(b)(b)(b)

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Calculate the volume of the solid obtained by rotating the region between the graph f (x) = x sin x and the x-axis, over [0, ] about: (a) the y-axis, x = 2 Stewart 396 (a)(a)(a)(a) (b)(b)(b)(b)

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Calculate the volume of the solid obtained by rotating the region between the graph f (x) = sin x and: (a) the x-axis, over [0, ] about the line x = 5 (b) t t t the line y = 4, over [0, 4], about the y-axis (b)(b)(b)(b) (a)(a)(a)(a)

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 E§ EE§ Example. Calculate the volume of the solid obtained by rotating the circle (x – 2)2 + y2 = 1 about the y-axis

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AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 Practice. Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, 9, 11, 14, 15, 17, 19 Suggested Extra Practice : pp 463-466, prob. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

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§ The Shell Method AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 The volume of the solid obtained by revolving the area under function h, can be calculated in the next form: Vertical axis of revolution Horizontal axis of revolution p(x) = distance to axis p(y) = distance to axis

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§ The Shell Method AP Calculus Volume: The Shell Method Larson – Hostetler – Edwards: Chapter 7.3 The volume of the solid obtained by revolving the area under function h, can be calculated in the next form: Vertical axis of revolution Horizontal axis of revolution p(y) = distance to axis p(x) = distance to axis

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6.3 Volumes of Revolution Tues Dec 15 Do Now Find the volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections.

6.3 Volumes of Revolution Tues Dec 15 Do Now Find the volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections.

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