Question

In Exercises 29-32, find the volume of the solid described. Find the volume of the solid generated by revolving the region bounded by the parabola \(y=x^{2}\) and the line \(y=1\) about (a) the line \(y=1.\) (b) the line \(y=2\) (c) the line \(y=-1\)

Short answer

The volumes of the solids generated by revolving the region bounded by the parabola \(y=x^{2}\) and the line \(y=1\) about the line \(y=1\), \(y=2\), and \(y=-1\) can be found by setting up integrals \(\pi \int_{-1}^{1} (1-x^{2})^{2} dx\), \(\pi \int_{-1}^{1} (2-x^{2})^{2} dx + \pi \int_{- \sqrt {3}}^{ \sqrt {3}} dx \), and \(\pi \int_{-1}^{1} (1+x^{2})^{2} dx\), respectively, and then solving them.

Step-by-step solution

Understanding the Solid and Selecting a Method

We are rotating the region bounded by the parabola \(y = x^{2}\) and the line \(y=1\) about the line \(y=1\), \(y=2\), and \(y=-1\). The solid thus formed will be a solid of revolution. The most convenient method to solve this problem might be the disk/washer method as the cross-sections of the solid will appear to be disks or washers.

Setting up the Integrals for Different Cases

(a) For the line \(y=1\), our radius would be \((1-x^{2})\) and the thickness would be \(\delta y\). This implies that the volume \(V\) would be \(\pi \int_{-1}^{1} (1-x^{2})^{2} dx\). (b) For the line \(y=2\), our radius would be \((2-x^{2})\) for \(x \leq 1\) and \(1\) for \(x > 1\). The integral here is broken into two parts such that the volume \(V\) would be \(\pi \int_{-1}^{1} (2-x^{2})^{2} dx + \pi \int_{- \sqrt {3}}^{ \sqrt {3}} dx \). (c) For line \(y=-1\), radius is \(1+x^{2}\) and thus volume \(V\) would be \(\pi \int_{-1}^{1} (1+x^{2})^{2} dx\).

Computing the integrals

Now the given integrals have to be computed which will yield the volume of the solid in each case. The integrals might be computed either by using basic properties of integrals or using a suitable substitution, if needed. This step will involve carrying out the details of the integrals which is basically the final calculations to get the volumes.