Finding volume bounded by given curves
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18. Compute the volume of the solid formed by revolving the region bounded by about (a) the x-axis; (b) y = 4.
20. Compute the volume of the solid formed by revolving the region bounded by and about (a) the y-axis; (b) x = 1.
4. Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by and revolved about .
6. Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by and , revolved about x = 2.
8. Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by , revolved about y = 4.
12. Use cylindrical shells to compute the volume of the region bounded by and x = 4, revolved about y = 2.
14. Compute the arc length exactly.
30. Set up the integral for the surface area of the surface of revolution, and approximate the integral with a numerical method.
revolved about the x-axis
36. Set up the integral for the surface area of the surface of revolution, and approximate the integral with a numerical method.
revolved about the x-axis
20. Evaluate the integral
14. Evaluate the integral x dx
Determine whether or not the integral is improper. If it is improper, explain why.
(see attached)
16. Find the indicated limit
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The solution gives detailed steps on ten calculus questions: finding volume bounded by given curves using different methods, calculating arc length and surface area.
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18. Compute the volume of the solid formed by revolving the region bounded by about (a) the x-axis; (b) y = 4.
(a)
(b)
20. Compute the volume of the solid formed by revolving the region bounded by and about (a) the y-axis; (b) x = 1.
(a)
(b)
4. Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by and revolved about .
The sketch is
Since the part above x-axis is symmetry with the part below x-axis, we just consider the part ...
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