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Computing Volumes of Solids Using Definite Integrals and Revolution Techniques

This section explores the application of definite integrals in computing the volumes of solids generated by revolving specific regions around axes. It covers the identification of normal lines to graphs, the continuity of derivatives of functions, and methods such as u-substitution for integration. Students will learn to find volumes of solids of revolution, particularly in cases involving shapes defined by functions in the first quadrant. The material also addresses various solid configurations and the impact of certain variables on volumes.

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Computing Volumes of Solids Using Definite Integrals and Revolution Techniques

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  1. Section 5.3 - Volumes by Slicing 7.3 I can use the definite integral to compute the volume of certain solids. Day 2: • 1. 02111a-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

  2. rotating region rotating region rotating region

  3. Find the volume of the solid generated by revolving the regions bounded by about the x-axis.

  4. Find the volume of the solid generated by revolving the regions bounded by about the x-axis.

  5. Find the volume of the solid generated by revolving the regions about the y-axis. bounded by

  6. Find the volume of the solid generated by revolving the regions bounded by about the x-axis.

  7. Find the volume of the solid generated by revolving the regions bounded by about the line y = -1.

  8. 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?

  9. Let R be the first quadrant region enclosed by the graph of a) Find the area of R in terms of k.

  10. 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.

  11. Let R be the first quadrant region enclosed by the graph of c) What is the volume in part (b) as k approaches infinity?

  12. 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.

  13. Let R be the region in the first quadrant under the graph of a) Find the area of R.

  14. 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

  15. 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

  16. 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.

  17. CALCULATOR REQUIRED

  18. 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.

  19. 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.

  20. 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) 2.79 b) 2.82 c) 2.85 d) 2.88 e) 2.91

  21. 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

  22. CALCULATOR REQUIRED

  23. NO CALCULATOR

  24. CALCULATOR REQUIRED

  25. NO CALCULATOR

  26. CALCULATOR REQUIRED

  27. CALCULATOR REQUIRED

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