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6.2 - Volumes

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Definition: Right Cylinder Let B 1 and B 2 be two congruent bases. A cylinder is the points on the line segments perpendicular to the bases that join B 1 and B 2. B1B1 B2B2 Alternate Definition A cylinder is a surface whose cross section in every plane parallel to a given plane are the same.

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Volumes of Solids P x is perpendicular to the x axis. A(x) is a function that represents the cross-sectional area where a ≤ x ≤ b.

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Volumes of Solids Establish n subintervals on [a, b] and let Δx. Choose the midpoint to be x i. Then a cylinder is formed with volume V(S i ) = A(x i ) Δx

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Estimate of the volume of the figure using n = 7 subintervals. The volume can be approximated by

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Definition of Volume Let S be a solid that lies between x = a and x = b. If the cross- sectional area of S in the plane P x, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is V = Area × Width

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Volumes of Solids of Revolution: 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 the y-axis. 2-D Graph What is the function that represents the radius? 3-D Graph What is the function that represents the area of the disk?

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It will be easier to integrate along the y-axis since it is perpendicular to the disks. Thus, the radius, x, is the distance from the y-axis to the function or The area of the disk is given by The volume of revolution is given by

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Volumes of Solids of Revolution: Example Let R be the region bounded by the graphs of y = x 2, x = 0, and y = 1. Compute the volume of the solid formed by revolving R about the x- axis. 2-D Graph What is the function that represents the radius? 3-D Graph What is the function that represents the area of the washer?

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Now we need to determine the area function for the washer.

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The volume can now be found by

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Volumes of Solids of Revolution: Example Let R be the region bounded by the graphs of y = x 2, x = 0, and y = 1. Compute the volume of the solid formed by revolving R about the line y = 2. 2-D Graph What is the function that represents the radius? 3-D Graph What is the function that represents the area of the washer?

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Now we need to determine the area function for the washer. The volume is given by

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