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SECTION 7.3 Volume

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VOLUMES OF AN OBJECT WITH A KNOWN CROSS-SECTION Think of the formula for the volume of a prism: V = Bh. The base is a cross-section of the prism. Now imagine that the shape has a very tiny thickness, thus forming a solid. We can think of the prism as a stack of many many of these solids. We can use calculus to find the volume of a solid by writing an expression for the area of a cross-section and integrating that expression over the length of the object.

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GENERAL FORMULA

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ANIMATION

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EXAMPLE 1

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EXAMPLE 2 Find the volume created by a solid whose base is the region y = x 2 for 0 < x < 3 if the cross sections are A) semicircles B) Isosceles right triangles with the leg on the base. Each cross section is perpendicular to the x-axis.

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EXAMPLE 3 Find the volume of an object that has a circular base of radius 2cm and a cross section that is perpendicular to the x-axis and is a right isosceles triangle with the leg on the base.

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VOLUME OF REVOLUTION

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DISC METHOD

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EXAMPLE

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WASHER METHOD

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WASHER METHOD CONTINUED

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EXAMPLE

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ROTATING ABOUT A LINE OTHER THAN THE X OR Y AXIS When rotating around an axis, the value of a function (biggie or smalls) tells you how far the function is from the axis. When rotating around a line other than an axis, you must write an expression that represents the distance from the curve(s) to that line.

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EXAMPLE Let R be the region bounded by y = 4 – x 2 and y = 0. Find the volume of the solids obtained by revolving R about each of the following…. (a) the x axis (b) the line y = -3 (c) the line y = 7 (d) the line x = 3

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