help? method of rings?

Question

Answer 1

(1)

Step 1:

The curve equations are , and -axis.

Method of Rings :

The volume of the solid obtained by rotating about -axis, the region of the curve  from  to is

.

Here .

Substitute in the above.

The limits of integration are and .

The volume of the solid is .

Solution :

The volume of the solid is .

Answer 2

(2)

Step 1 :

The curve equations are , and -axis.

Method of Rings :

The volume of the solid obtained by rotating about -axis, the region of the curve  from  to is

.

Here the curve is .

Find the value of x for .

The limits of integration are .

The volume of the solid is cubic units.

Solution :

The volume of the solid is cubic units.

Answer 3

(3)

Step 1:

The curve equations are , and -axis.

Method of Rings :

The volume of the solid obtained by rotating about -axis, the region of the curve  from  to is

.

Here .

The line is .

The limits of integration are x = 0 and x = 8.

The volume of the solid is cubic units.

Solution :

The volume of the solid is cubic units.

Answer 4

(4)

Step 1:

The curve equation is , and -axis.

Method of Rings :

The volume of the solid obtained by rotating about -axis, the region of the curve  from  to is

.

Here the curve is .

The line .

The limits of integration are y = 0 and y = 8.

The volume of the solid is cubic units.

Solution :

The volume of the solid is cubic units.