Question

Given the region bounded by the graphs of \(y=x \sin x, y=0, x=0,\) and \(x=\pi,\) find (a) the volume of the solid generated by revolving the region about the \(x\) -axis. (b) the volume of the solid generated by revolving the region about the \(y\) -axis. (c) the centroid of the region.

Short answer

The volume of the solid generated by revolving the region about the x-axis and the y-axis, as well as the coordinates of the centroid would be obtained by numerically solving the integrals set up in steps 1, 2 and 3 respectively. Due to the complexity of these integrals, a closed form solution is not simple to obtain and thus numerical solutions are preferred.

Step-by-step solution

Volume of solid around x-axis

First, calculate the volume of the solid generated by revolving the region about the x-axis. This uses the disc method. The generic formula is given by: \( V = \pi \int_{a}^{b} [f(x)]^2 dx \). Substituting \(f(x)=x \sin x\) gives: \( V_1 = \pi \int_{0}^{\pi} [x \sin x]^2 dx \). Solve this integral numerically to obtain the volume.

Volume of solid around y-axis

Next, calculate the volume of the solid generated by revolving the region about the y-axis. This involves the cylindrical shell method. Solving for x in the equation \(y=x \sin x\) is nontrivial, thus a numerical method is preferred to solve this part. The volume is given as \(V_2 = 2 \pi \int_0^{b} [x \cdot f(x)] dx\). Substituting \(f(x)=x \sin x\) gives \(V_2 = 2 \pi \int_0^\pi [x \cdot x \sin x] dx\). Numerically solve this integral to obtain the volume.

Centroid of the region

Finally, calculate the centroid of the region. The centroid coordinates are given by \((\overline{x}, \overline{y})\), where \(\overline{x} = \frac{1}{A} \cdot \int_{a}^{b} [x \cdot f(x)] dx\) and \(\overline{y} = \frac{1}{2A} \cdot \int_{a}^{b} [f(x)]^2 dx\). The area A in this case is: \(A = \int_0^\pi [x \sin x] dx\). These integrals yield the \(x\) and \(y\) coordinates of the centroid when computed numerically, where \(\overline{x}, \overline{y}\) are centroid on the x and y-axis, respectively.