Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=x^{2}+1, \quad y=-x^{2}+2 x+5, \quad x=0, \quad x=3 $$

Short answer

The volume of the solid figure of revolution is obtained by first graphing the equations to understand the bounded region, then setting up and evaluating the integral that represents the volume of the solid.

Step-by-step solution

Visualising and Understanding the Graphs and Bounded Region

The equations \(y=x^{2}+1\) and \(y=-x^{2}+2x+5\) should be graphed. Their intersection points, which become the limits of integration, should be identified. In addition, the bounded region between \(x=0\) and \(x=3\) should be recognized.

Finding the Intersection Points

To find the points of intersection between the two curves, set the equations equal to each other: \(x^{2}+1=-x^{2}+2x+5\). Solving this will give the x values of the intersection points.

Setting up the Volume Integral

Next, the formula for the volume of a solid of revolution using the method of disks is used, which is \(V = \pi \int_{a}^{b} [R(x)]^2 dx\), where \(R(x)\) is the radius of the disk at x. In this case, the outer radius, \(R(x)\), is defined by the equation \(y = -x^{2} + 2x + 5\) while the inner radius, \(r(x)\), is defined by \(y = x^{2} + 1\). Therefore, the integral becomes \(V = \pi \int_{a}^{b} [(R(x))^2 - (r(x))^2] dx\).

Evaluating the Integral

Evaluate the integral over the defined limits to find the volume of the solid.