find the volume of solid of revolution formed by roation...

Question

1a. Find the volume of solid of revolution formed by rotating about x−axis the area bounded by y = f(x) = x^(1/3), x-axis, x = 0 and x = 8.

1b. Find the volume of solid of revolution formed by rotating about x−axis the area bounded by y = f(x) = e^x + 2, x-axis, x = −2 and x = 1. (Final answer may be kept in terms of various powers of ”e”.)

2a. Find the volume of solid of revolution formed by rotating about x−axis the area bounded by y = f(x) = 3x + 2, x-axis, x = 0 and x = 1.

2b. Find the volume of solid of revolution formed by rotating about x−axis the area bounded by y = f(x) = sqrt(3x^2 + 6x + 7), x-axis, x = −3 and x = 1

3a. Find the average value of f(x) = lnx on [1, e]. (Final answer may be kept in terms of various powers of ”e”.)

3b. Find the average value of f(x) = (x + 1)/( sqrt(x)) on [1, 9].

Answer 1

(1a)

Step 1:

The equations are  , -axis, , and about -axis.

Disk method :

The volume of the solid  is , where  is the cross sectional area of the solid .

.

Graph:

Graph the equations are , -axis, , .

Shade the required region.

Observe the graph:

Radius .

Integral limits are  and  .

Step 2:

Power rule of integration: .

.

Volume of the solid is  cubic units.

Solution:

Volume of the solid is cubic units.

Answer 2

(1b)

Step 1:

The equations are  , -axis, , and about -axis.

Disk method :

The volume of the solid  is , where  is the cross sectional area of the solid .

.

Graph:

Graph the equations are , -axis, , .

Shade the required region.

Radius is 

Integral limits are  and .

Step 2:

Find the volume of the solid.

Substitute corresponding values in the volume formula.

Integral formula: .

 cubic units.

Volume of the solid is cubic units.

Solution:

Volume of the solid is  cubic units.

Answer 3

2(a)

Step 1:

The equations are , -axis, and .

The volume of the solid generated revolving about the -axis.

Washer method:

Volume of the solid: .

Where is the outer radius, is the inner radius.

Graph the functions , , and .

Observe the graph:

The outer radius is .

The inner radius is .

The integral limits are and .

Step 2:

Find the volume of the solid.

Substitute , , and in .

Apply power rule of integration: .

.

The volume of the solid is .

Solution:

The volume of the solid is .

Answer 4

2(b)

Step 1:

The equations are , -axis, and .

The volume of the solid generated revolving about the -axis.

Washer method:

Volume of the solid: .

Where is the outer radius, is the inner radius.

Graph:

Graph the functions , , and .

Observe the graph:

The outer radius is .

The inner radius is .

The integral limits are and .

Step 2:

Find the volume of the solid.

Substitute , , and in

Apply power rule of integration: .

.

The volume of the solid is .

Solution:

The volume of the solid is .

Answer 5

3(a)

The function is  on interval .

Average value of the function  on is defined as .

Here, and .

Substitute , and in .

The average value is

Apply formula: .

.

The average value is .

Answer 6

3(b)

The function is  and the inerval is .

Average value of the function on is defined as .

Here  .

Average value of  is 

Power rule of integration: .

.

Average value of the function  on  is .