Question

Find the volume ( V ) using the disk method of a solid obtained by rotating the region bounded by ( 4x² = 3y ), ( x = 0 ), and ( y = 4 ). Evaluate the radius ( r ) of rotation at point ( y = 2.4 ).

(A) ( V = )
(B) ( r(y) = )

Answer 1

To find the volume using the disk method, integrate the area of the cross-section. The radius of each disk is the value of x at that point. Substituting y into the equation for x gives the radius at a specific y value.

Step-by-step explanation:

The volume of the solid obtained by rotating the region bounded by (4x² = 3y), (x = 0), and (y = 4) can be found using the disk method. The disk method involves integrating the areas of small disks to find the total volume.

To find the volume using the disk method, we need to integrate the area of the cross-section from x = 0 to the point where the curve 4x² = 3y intersects the line y = 4. The radius of each disk is the value of x at that point.

Since the curve is given as 4x² = 3y, we can rewrite it as x = sqrt(3/4 * y). To find the radius at a specific y value, substitute y into the equation for x and calculate the value.

For example, to find the radius at y = 2.4, substitute y = 2.4 into x = sqrt(3/4 * y) to get x = sqrt(3/4 * 2.4) = sqrt(3.6). Therefore, the radius at y = 2.4 is approximately 1.897 units.