Determining the Volume of Solid of Revolution: A Case Study

Volume of solid of revolution is a classic problem in A-Level Mathematics, showcasing the washer method. This article will demonstrate how to solve a problem involving the region bounded by the curve y 4 - x2 and the line y x2, when this region is revolved around the x-axis. By understanding the difference between the outer and inner radii, we can apply the washer method to find the volume of the resulting solid.

Setting Up the Problem

We need to determine the volume of the solid generated by revolving the region bounded by the curve y 4 - x2 and the line y x2 about the x-axis. The first step is to find the points of intersection of these two curves:

Intersection Points

x2 4 - x2 2x2 4 x2 2 x ±sqrt{2}

Thus, the points of intersection are x -sqrt{2} and x sqrt{2}. However, for the context of this example, let's consider the interval x -2 to x 1.

Applying the Washer Method

The washer method is a technique in calculus used for calculating the volume of a solid of revolution. The formula is:

( V pi int_{a}^{b} (R^2 - r^2),dx )

Where:

R is the outer radius, r is the inner radius, a and b are the bounds of integration.

Calculating the Limits and Radii

First, we determine the limits of integration by finding the points of intersection. The region of interest is bounded between x -2 and x 1 because these are the points where the curve and the line intersect. We can see that:

For any x in the interval [-2, 1], 4 - x2 is greater than x2. This can be verified by evaluating the function at x 0: 4 - 02 4 02 0

Thus, the outer radius R is given by 4 - x2, and the inner radius r is given by x2.

Setting Up the Integral

The integral for the volume can be set up as follows:

( V pi int_{-2}^{1} [(4 - x2)2 - (x2)2],dx )

Simplifying the integrand:

( V pi int_{-2}^{1} [(16 - 8x2 x?) - x?],dx )

Further simplification:

( V pi int_{-2}^{1} (16 - 8x2),dx )

Now, we integrate term-by-term:

( V pi int_{-2}^{1} 16,dx - 8pi int_{-2}^{1} x2,dx )

Evaluating the integrals:

( V pi [16x]_{-2}^{1} - frac{8pi}{3} [x3]_{-2}^{1} )

Substituting the bounds:

( V pi (16 - (-16)) - frac{8pi}{3} [1 - (-8)] )

Simplifying:

( V 32pi - frac{8pi}{3} cdot 9 )

Further simplification:

( V 32pi - 24pi )

Finally:

( V 8pi )

Therefore, the volume of the solid of revolution is boxed{frac{108pi}{5}}.

Conclusion

Using the washer method, we can effectively determine the volume of solids of revolution. By carefully setting up the limits of integration and the radii, we can solve complex problems in A-Level Mathematics. The key steps are finding the points of intersection, determining the radii, and setting up and evaluating the integral.