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7.3 Volumes

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Quick Review

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What you’ll learn about Volumes As an Integral Square Cross Sections Circular Cross Sections Cylindrical Shells Other Cross Sections Essential Question How can we use calculus to compute volumes of certain solids in three dimensions? http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html http://www.math.psu.edu/dlittle/java/calculus/volumedisks.html

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Find the volume of the solid when the curve is rotated around the x-axis.

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Volume of a Solid The definition of a solid of unknown integrable cross section area A(x) from x = a to x = b is the integral of A from a to b, How to Find Volumes by the Method of Slicing 1.Sketch the solid and a typical cross section. 2.Find a formula for A(x). 3.Find the limits of integration. 4.Integrate A(x) to find the volume.

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Example Square Cross Sections 1.A pyramid 3 m high has congruent triangular sides and a square base that is 3 m on each side. Each cross section of the pyramid parallel to the base is a square. Find the volume of the pyramid. 1. Sketch: Draw the pyramid with its vertex at the origin and its altitude along the interval 0 < x < 3. Sketch a typical cross section at a point x between 0 and 3. 2. Find a formula for A(x): The cross section at x is a square x meters on a side, so the formula will be:

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Example Square Cross Sections 1.A pyramid 3 m high has congruent triangular sides and a square base that is 3 m on each side. Each cross section of the pyramid parallel to the base is a square. Find the volume of the pyramid. 3. Find the limits of integration: The square goes from x = 0 to x = 3. 4. Integrate to find the volume: m 3

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Example A Solid of Revolution 2.The region between the graph f (x) = 2 + x cos x and the x-axis over the interval [– 2, 2] is revolved about the x-axis to generate a solid. Find the volume of the solid. http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html Revolving the region about the x-axis generates a ____________ solid. vase-shaped The cross section at a typical point x is __________. circular The radius is equal to ______. f (x)

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Example Finding Volumes Using Cylindrical Shells 3.Find the volume of the solid generated by revolving about the x-axis the region bounded by http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html

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Pg. 406, 7.3 #1-25 odd

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Cylindrical Shell Method Use the shell method when the axis of revolution is perpendicular to the axis containing the natural interval of integration. Instead of summing volumes of thin slices, we sum volumes of thin cylindrical shells that grow outward from the axis of revolution. 4.The region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid.

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Example Finding Volumes Using Cylindrical Shells 5.The region bounded by the curve y =, y = x, and x = 0 is revolved about the y-axis to form a solid. Use cylindrical shells to find the volume of the solid.

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Example Other Cross Sections 6.A solid is made so that its base is the shape of the region between the x-axis and one arch of the curve y = 2 sin x. Each cross section cut perpendicular to the x-axis is a semicircle whose diameter runs from the x-axis to the curve. Find the volume of the solid. Radius of the semicircle is

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Pg. 251, 4.6 #1-35 odd

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Quick Quiz Sections 7.1-7.3

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