Question

The region in the first quadrant bounded by the x-axis, the line x = In(pi), and the curve y = sin(ex) is rotated about the x-axis. What is the volume of the generated solid? (5 points)

0.906
0.795
2.846
2.498​

Answer 1

D

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:

`\displaystyle V=\pi\int_a^bR(x)^2\, dx`

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:

`\displaystyle V=\pi\int_0^(\ln(\pi))(\sin(e^x))^2\, dx`

Simplify:

`\displaystyle V=\pi\int_0^(\ln(\pi))\sin^2(e^x)\, dx`

Approximate. So, the volume of the generated solid is:

`V\approx 2.498\text{ units}^3`

The solid is shown in the second figure.

(Courtesy of WolframAlpha.)