Question

Use shells to find the volumes of the following solids: the rotated regions lie between the curve and the x-axis and are rotated about the y-axis; (a) y=5x 3 ,x=0 and x=2; (b) y=(1+x 2 ) 3 ,x=0 and x=1; (c) y= x 2 1 ​ ,x=1 and x=100.

Answer 1

The volume of a solid obtained by rotating a curve around the y-axis using shells can be found using the formula V = 2∫(radius)(height) dx.

This correct answer is none of the above.

Step-by-step explanation:

To find the volume of the solid obtained by rotating a curve around the y-axis using shells, we can use the formula:

V = 2π∫(radius)(height) dx

where the radius is the distance from the y-axis to the curve, and the height is the infinitesimal thickness of the shell.

(a) For y = 5x^3, x = 0, and x = 2:

The radius is x, and the height is 5x^3 - 0.

Thus, the volume is:

V = 2π∫x(5x^3 - 0) dx

(b) For y = (1 + x^2)^3, x = 0, and x = 1:

The radius is x, and the height is (1 + x^2)^3 - 0.

Thus, the volume is:

V = 2π∫x((1 + x^2)^3 - 0) dx

(c) For y = x^2 + 1, x = 1, and x = 100:

The radius is x - 1, and the height is (x^2 + 1) - 0.

Thus, the volume is:

V = 2π∫(x-1)((x^2 + 1) - 0) dx

This correct answer is none of the above.