7_2_Volumes_of_Revolution_Disk - 7.2 Volumes of Revolution the Disk Method HW 7.2 115 1927 35 63(calc StepbyStep Procedure Practice in Problem Setup

7_2_Volumes_of_Revolution_Disk - 7.2 Volumes of Revolution...

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Unformatted text preview: 7.2 Volumes of Revolution: the Disk Method HW: 7.2 # 115, 1927, 35, 63 (calc) StepbyStep Procedure Practice in Problem Setup Calculus Home Page Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 1 7.2 Volumes of Revolution: the Disk Method From geometry, we find volumes of readily defined geometric figures. For example: Geometric Figure Sphere Right Circular Cone Right Circular Cylinder Volume V = 4 r3 3 V = 1 r2 h 3 V = r2 h Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 2 7.2 Volumes of Revolution: the Disk Method To begin: Focus on: Right Circular Cylinder Volume V = r2 h Consider a soda can, and flip it onto it's side. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 3 7.2 Volumes of Revolution: the Disk Method y = 3 thickness: x =2.01.9= 0.1 can be different curves Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 4 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 5 7.2 Volumes of Revolution: the Disk Method Known geometrical figures can use a formula, or calculus. For nonstandard geometrical figures, there is no formula. Instead, use calculus. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 6 7.2 Volumes of Revolution: the Disk Method compare to formula for cone Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 7 G. Battaly 2011 8 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 9 7.2 Volumes of Revolution: the Disk Method start Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 10 7.2 Volumes of Revolution: the Disk/Washer Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 11 G. Battaly 2011 12 7.2 Volumes of Revolution: the Disk Method finish Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 13 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 14 7.2 Volumes of Revolution: the Disk Method Rotate around the yaxis (or a vertical line) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 15 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 16 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 17 7.2 Volumes of Revolution: the Disk Method Volumes of Revolution Disk Method 1. Sketch the curves and identify the region, using the points of intersection. 2. Locate the axis of revolution on the sketch. 3. Decide whether to use a horizontal or vertical rectangle. The rectangle should be perpendicular to the axis of revolution. 4. Sketch the rectangle and determine the variable of integration. If the rectangle is horizontal, then integrate with respect to y (use dy). The integrand must be in terms of y. If the rectangle is vertical, then integrate with respect to x (use dx). The integrand must be in terms of x. 5. Determine the integrand: R2, or R2 r2 ? a) If the rectangle touches the axis of revolution, identify R as the length of the rectangle. Find R in terms of the appropriate variable (see above), and use R2 as the integrand. b) If the rectangle does not touch the axis of revolution, identify R as the distance of the furthest end of the rectangle from the axis of revolution and r as the distance of the closest end of the rectangle from the axis of revolution. Use R2 r2 as the integrand. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 18 7.2 Volumes of Revolution: the Disk Method Find the volume of the solids generated by revolving the region bounded by: y = 2 x 2 y = 0 x = 2 about the given axes. a) yaxis b) xaxis c) y = 8 d) x = 2 1st) sketch Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 19 7.2 Volumes of Revolution: the Disk Method link link link link 2nd) axis of rev a) yaxis b) xaxis c) y = 8 d) x = 2 vertical horizontal horizontal vertical 3rd) ref. rectangle: _|_ to axis rev Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 20 7.2 Volumes of Revolution: the Disk Method 2nd) axis of rev a) yaxis vertical 3rd) ref. rectangle: _|_ to axis rev 4th) decide: dy or dx? y=dy y=dy ( c d )dy ref. rectangle does NOT touch the axis of revol. Use R2 r2 5th) decide: R2 or R2 r2 ? return to problem Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 21 7.2 Volumes of Revolution: the Disk Method 2nd) axis of rev b) xaxis horizontal 3rd) ref. rectangle: _|_ to axis rev 4th) decide: dy or dx? x=dx ( a b )dx ref. rectangle touches the axis of revol. Use R2 x=dx 5th) decide: R2 or R2 r2 ? return to problem Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 22 7.2 Volumes of Revolution: the Disk Method 2nd) axis of rev c) y = 8 horizontal 3rd) ref. rectangle: _|_ to axis rev 4th) decide: dy or dx? x=dx ( a b )dx ref. rectangle does NOT touch the axis of revol. Use R2 r2 x=dx 5th) decide: R2 or R2 r2 ? return to problem Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 23 7.2 Volumes of Revolution: the Disk Method 2nd) axis of rev d) x = 2 vertical 3rd) ref. rectangle: _|_ to axis rev 4th) decide: dy or dx? y=dy y=dy ( c d )dy ref. rectangle does touches the axis of revol. Use R2 return to problem 5th) decide: R2 or R2 r2 ? Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 24 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 25 7.2 Volumes of Revolution: the Disk Method Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Homework Part 3 G. Battaly 2011 26 G. Battaly 2011 27 G. Battaly 2011 28 G. Battaly 2011 29 G. Battaly 2011 30 ...
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