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AP CALCULUS - AB Section Number: LECTURE NOTES Topics: Volume: The Disk & Washer Methods MR. RECORD Day: 1 of 2 7.2 Area is only one of the many applications of the definite integral. We can also use it to find the volume of a threedimensional solid. The solids which you will focus on in this unit are those with cross sections that are all similar. w w R( x ) R( x ) A 2-dimensional figure ………..which, when rotated about the x-axis becomes a…….3-dimensional figure Recall: The formula for the volume of a cylinder is V r h . A disk is basically a very short cylinder where the height, h is actually the w value. The radius retains its actual meaning. The pops out in front and you get the following formula below: 2 The Disk Method To find the volume of a solid of revolution with the disk method, use one of the following: Horizontal Axis of Revolution b Vertical Axis of Revolution d Volume V R(x) dx Volume V R(y) dy 2 a 2 c x d y R( x ) a b c R(y) Example 1: Using the Disk Method. Find the volume of the solid formed by revolving the region bounded by the graphs of f (x) sin x and the x-axis about the x-axis (0 x ) . Sketch the graph and draw a representative rectangle. y x Example 2: Revolving About a Line That Is Not A Coordinate Axis. Find the volume of the solid formed by revolving the region bounded by the graphs of f (x) 2 x 2 and g(x) 1 about the line y 1 Sketch the graph and draw a representative rectangle. y x The Washer Method To find the volume of a solid of revolution with the washer method, use one of the following: Horizontal Axis of Revolution b Vertical Axis of Revolution d Volume V R(x) r (x) dx 2 2 Volume V R(y) r (y) dy a 2 2 c R(y) d R( x ) r (x) a b c r (x) Example 3: Using the Washer Method. Find the volume of the solid formed by revolving the region bounded by the graphs of f (x) x and y x 2 about the x-axis. Sketch the graph and draw a representative rectangle. y x Example 4: Integrating with Respect to y, Two Integral Case. Find the volume of the solid formed by revolving the region bounded by the graphs of y x 2 1, y 0, x 0, and x 1 about the y-axis. Sketch the graph and draw a representative rectangle. y x Think of the above example in some other ways. What would your integration set-up look like if you were to revolve the region about (a) the line x =–1? (b) the line y = -2? (c) the line x = 2? Advice 1. Draw a representative rectangle that begins from the axis of revolution to the farthest boundary. (This distance will represent R) 2. Draw a representative rectangle that begins from the axis of revolution to the nearest boundary. (This distance will represent r) AP CALCULUS Section Number: LECTURE NOTES Topics: Volume: Using KNOWN CROSS SECTIONS MR. RECORD Day: 2 of 2 7.2 Solids With Known Cross Sections Volumes of Solids with Known Cross Sections 1. For cross sections of area A(x) taken perpendicular to the x-axis. b Volume A( x) dx a 2. For cross sections of area A(y) taken perpendicular to the y-axis. d Volume A(y) dy c Example 5: Triangular Cross Sections. Find the volume of the solid in which the base is the region bounded by the graphs of x x f (x) 1 , g(x) 1 , and x 0. 2 2 a. The cross sections perpendicular to the x-axis are squares. y x b. The cross sections perpendicular to the x-axis are semi-circles. c. The cross sections perpendicular to the x-axis are equilateral triangles.