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**Section 5.3 - Volumes by Slicing**

I can use the definite integral to compute the volume of certain solids. Day 2: a-dLet f be the function defined by Write an equation of the line normal to the graph of f at x = 1. b. For what values of x is the derivative of f, f ‘ (x), not continuous? Justify your answer. c. Determine the limit of the derivative at each point of discontinuity found in part (b). d. Can be completed using the method of u-substitution? If yes, complete the integration. If no, explain why u-substitution cannot be used for Solids of Revolution 7.3

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rotating region rotating region rotating region

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**Find the volume of the solid generated by revolving the regions**

bounded by about the x-axis.

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**Find the volume of the solid generated by revolving the regions**

bounded by about the x-axis.

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**Find the volume of the solid generated by revolving the regions**

bounded by about the y-axis.

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**Find the volume of the solid generated by revolving the regions**

bounded by about the x-axis.

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**Find the volume of the solid generated by revolving the regions**

bounded by about the line y = -1.

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NO CALCULATOR Let R be the first quadrant region enclosed by the graph of a) Find the area of R in terms of k. Find the volume of the solid generated when R is rotated about the x-axis in terms of k. c) What is the volume in part (b) as k approaches infinity?

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**Let R be the first quadrant region enclosed by the graph of**

a) Find the area of R in terms of k.

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**Let R be the first quadrant region enclosed by the graph of**

Find the volume of the solid generated when R is rotated about the x-axis in terms of k.

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**Let R be the first quadrant region enclosed by the graph of**

c) What is the volume in part (b) as k approaches infinity?

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CALCULATOR REQUIRED Let R be the region in the first quadrant under the graph of a) Find the area of R. The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k? Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.

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**Let R be the region in the first quadrant under the graph of**

a) Find the area of R.

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**Let R be the region in the first quadrant under the graph of**

The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k? A

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**Let R be the region in the first quadrant under the graph of**

Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares. Cross Sections

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**The base of a solid is the circle . Each section of the**

solid cut by a plane perpendicular to the x-axis is a square with one edge in the base of the solid. Find the volume of the solid in terms of a.

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CALCULATOR REQUIRED

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**Let R be the region marked in the first quadrant enclosed by**

the y-axis and the graphs of as shown in the figure below Setup but do not evaluate the integral representing the volume of the solid generated when R is revolved around the x-axis. R Setup, but do not evaluate the integral representing the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares.

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**Let R be the region in the first quadrant bounded above by the**

graph of f(x) = 3 cos x and below by the graph of Setup, but do not evaluate, an integral expression in terms of a single variable for the volume of the solid generated when R is revolved about the x-axis. Let the base of a solid be the region R. If all cross sections perpendicular to the x-axis are equilateral triangles, setup, but do not evaluate, an integral expression of a single variable for the volume of the solid.

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**The volume of the solid generated by revolving the first quadrant**

region bounded by the curve and the lines x = ln 3 and y = 1 about the x-axis is closest to a) b) c) d) e) 2.91

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**The base of a solid is a right triangle whose perpendicular sides**

have lengths 6 and 4. Each plane section of the solid perpendicular to the side of length 6 is a semicircle whose diameter lies in the plane of the triangle. The volume of the solid in cubic units is: a) 2pi b) 4pi c) 8pi d) 16pi e) 24pi

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CALCULATOR REQUIRED

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NO CALCULATOR

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CALCULATOR REQUIRED

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NO CALCULATOR

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CALCULATOR REQUIRED

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CALCULATOR REQUIRED

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Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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