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Volume Calculations for Revolved Solids with Different Orientations

Explore volume calculations of solids generated by revolving regions about different axes and lines. Calculate volumes and areas in various scenarios. Understand changing volumes as parameters vary. Utilize calculus principles for intricate solid shapes. No calculator needed for specific calculations, but calculations involving specific shapes may require a calculator. Enhance your understanding of solid geometry through these exercises.

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Volume Calculations for Revolved Solids with Different Orientations

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  1. 7.3 Volumes by Slicing Solids of Revolution

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

  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 about the y-axis. bounded by

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

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

  7. 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?

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

  9. 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.

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

  11. 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.

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

  13. 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

  14. 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

  15. 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. (NO PI)

  16. CALCULATOR REQUIRED

  17. 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.

  18. 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.

  19. 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

  20. 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

  21. CALCULATOR REQUIRED

  22. NO CALCULATOR

  23. CALCULATOR REQUIRED

  24. NO CALCULATOR

  25. CALCULATOR REQUIRED

  26. CALCULATOR REQUIRED

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