Find the volume of the solid revolved around y=x^3, y=0, x=2 about the x-axis

Answer 1

The first thing you need to determine is which direction you are going to integrate. Since our solid is formed by rotating about the x-axis, when we imagine the infinitely thin disks we will get, they will be oriented so that their radius measures in the y direction and their height (or thickness) measures in the x direction. Therefore, we will want to integrate with respect to x.

So when we set up our integral to add up all the volumes of all the disks, dx will represent the height of each cylinder and the radius will come from y=x³ and will be based on the cylinder volume equation V=pi*r²h.

∫₀² pi * (r)² h

∫₀² pi (x³)² dx

Then you should be able to simplify and evaluate this integral to get your answer.

Here is a similar problem explained with pictures and slightly more detail if you get stuck: https://jakesmathlessons.com/integrals/rotating-volumes-with-the-disk-method/

Answer 2

The volume has a circle cross section as shown. The volume is the integral of the area of the circle (πr²) times its thickness (dx) from 0 to 2.

Although the area of the circle is πr², the "r" is the y value of the function y = x³ so r = x³ meaning:

V = ₀∫²π(x³)²dx

V = ₀∫²πx⁶dx = π/7x⁷ evaluated between 0 and 2

V = π/7(2⁷) - π/7(0⁷)

V =(128/7)π

Answer 3

The volume of a solid revolved around the x-axis is:

V = π ∫_a^b [f(x)]² dx

Therefore, we have:

V = π ∫₀² x⁶ dx = (π/7)[x⁷]₀² = (π/7)(128) = 128π/7

Answer 4

1) Draw the region on 2-d

notice region is in quadrant 1. And since we are revolving around y axis

we need to write these two functions which bound the region in relation

1) y(x)=x^3

2) y = 0

Formulate expression TOP MINUS BOTTOM AKA DISH-WASHER METHOD

x^3 - 0

A(x) = pi( expression) ^2

A(y) = pi(x^3 - 0)^2

To find volume integrate A(y) where the bound a and b are y values of endpoint thus from [0 2] since they intersect at point (2,8).

thus

V = integral A(x) from 0 to 2 dx

V= pi * int (A(x),0,2,dx)

V= 128pi/7

V= 57.4462 units cubed

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-Maks