Answer:
D
Step-by-step explanation:
Please refer to the graph below.
So, we want to find the volume of the solid generated by revolving the green area about the x-axis.
We can use the disk method. The disk method is given by:
[tex]\displaystyle V=\pi\int_a^bR(x)^2\, dx[/tex]
Where R(x) is the radius or height of the representative rectangle.
We are integrating from x = 0 to x = ln(π). The height of a representative rectangle is given by y. Therefore, the volume is:
[tex]\displaystyle V=\pi\int_0^{\ln(\pi)}(\sin(e^x))^2\, dx[/tex]
Simplify:
[tex]\displaystyle V=\pi\int_0^{\ln(\pi)}\sin^2(e^x)\, dx[/tex]
Approximate. So, the volume of the generated solid is:
[tex]V\approx 2.498\text{ units}^3[/tex]
The solid is shown in the second figure.
(Courtesy of WolframAlpha.)
