The region in the first quadrant bounded by the x-axis, the line x = π, and the curve y = sin(sin(x)) is rotated about the x-axis. What is the volume of the generated solid?

Question

Question options:

1)	1.219
2)	3.830
3)	1.786
4)	5.612

Answer 1

1. To solve this first you need to make a graph of the function, from which you will see that the area in question is a semi-circle/semi-oval under the y function and had has x-intercepts at x=0, and x = π.
2. The question asks for the volume of the figure created by the semi-circle/semi-oval rotated around the x-axis and by doing a quick sketch of it at different angles around the x-axis you can see that if you were to slice the generated solid, it would have circle faces going along the axis.
3. These faces are what is going to help you find the volume. If you can find the area of each circle in the shape and add their infinite amount up, you could theoretically get the figure's volume. Practically this can be done through using a definite integral from 0 to π of the equation of a circle's area, area = πr² .In this particular instance r = y = sin(sin(x)) as the output of y determines how big r is.
4. Lastly plug in the integral π ∫₀^π (sin(sin(x))² dx into a graphing calculator or an online integral calculator and an approximation of 3.82995 will be generated. Note that it is not possible to do by hand as there is know way to calculate an anti-derivative by hand for this particular problem.
5. Thus option 2 is your answer.

Answer 2

Volume V = π∫₀^π(sin(sin(x)))²dx ≈ 3.830.

To evaluate this integral I use TI-84, MATH Menu, FntInt command.

Answer is 2): V ≈ 3.830

Answer 3