# Finding volume bounded by given curves

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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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 ...

#### Solution Summary

The solution gives detailed steps on ten calculus questions: finding volume bounded by given curves using different methods, calculating arc length and surface area.

Region R bounded by the given curves is revolved

(See attached file for full problem description)

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Find the volume of the solid generated when the region R bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps

A sketch the region R

B show a typical slice properly labeled

C write a formula for the approximate volume of the shell generated by this slice

D set up the corresponding integral

E Evaluate this integral

1. y= x^2 , x=I y=0 about the y axis

2. y=9 - x^2 ( x> 0 ), x=0 y=0

3. y=9-x^2 (x>0), x=0, y=o ; about the line x=3

4. y==x^2, y=3x ; about the y-axis

5. x = sqrt y + 1 y=4 , y=0 ; about the x-axis

6. x = sqrt 2y 1 , y= 2 x= 0 ; about the line y =3

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