Question

Find the volume of the solid generated by revolving the region

bounded by y=2(square root of sinx), y=0, and x(sub1)=pi/3,
x(sub2)=5pi/6 about the x axis. Round to the nearest
hundreth.

Answer 1

Explanation:

To find the volume of the solid generated by revolving the given region about the x-axis, we can use the method of cylindrical shells.

The region is bounded by the curves y=2sin⁡xy=2sinx

​, y=0y=0, x=π3x=3π​, and x=5π6x=65π​.

The volume of the solid can be calculated using the formula:

V=2π∫abx⋅f(x) dxV=2π∫ab​x⋅f(x)dx

Where aa and bb are the bounds of integration (in this case, π33π​ and 5π665π​), and f(x)f(x) is the height of the shell, which is given by f(x)=2sin⁡xf(x)=2sinx

​.

So, the volume can be calculated as:

V=2π∫π35π6x⋅2sin⁡x dxV=2π∫3π​65π​​x⋅2sinx

​dx

Before proceeding, let's simplify the integral by rearranging it:

V=4π∫π35π6xsin⁡x dxV=4π∫3π​65π​​xsinx

​dx

Now, use calculus techniques to evaluate this integral. Unfortunately, this integral does not have a simple closed-form solution. You'll likely need to use numerical integration methods or appropriate software to compute the value of the integral.

Once you've computed the integral, you'll have the volume of the solid generated by revolving the given region about the x-axis. Be sure to round the result to the nearest hundredth as requested.