91Ó°ÊÓ

The disk method is a technique used to find volumes of solids of revolution. It involves slicing the solid perpendicular to the axis of revolution into thin disks or cylinders. Each disk has a small thickness, which is infinitely thin, represented by \(abla x\) in calculus terms.
To apply the disk method, you revolve a region around an axis, such as the y-axis in the given problem. This converts the two-dimensional region defined by the curves into a three-dimensional solid.
The volume of each tiny disk is approximately \( \pi [f(x)]^2 abla x \), where \( f(x) \) is the radius of the disk at a particular x-value. Adding up (integrating) all these disk volumes across the range covers the entire solid.

Calculating the volume of a solid of revolution is central to understanding this type of calculus problem. The overall process involves determining how much space the solid occupies once the region is rotated 360 degrees around the given axis.
In the exercise, we use the disk method formula \( V = \pi \int_{a}^{b} [f(x)]^2 dx \). This represents summing up the volume of all infinitesimally thin disks from \(x = a\) to \(x = b\). The limits of integration (a and b) are determined by where the curves intersect.
In our case, we break it into two parts because the region changes its defining function at \(y = 0\). Volume is calculated separately for each segment, using the respective function to define the radius of each disk.

Finding intersection points is a critical first step when dealing with regions bounded by curves. These are the x-values where the given curves meet or cross one another, forming the boundaries of the region you're revolving.
To find intersection points in this problem, we equate the functions \(x^2 = 2 - x\) and solve the resulting quadratic equation.
Solving \(x^2 + x - 2 = 0\) gives us the roots \(x = -2\) and \(x = 1\). These x-values are used as the limits of integration when setting up the integrals in the disk method.
Intersection points help define the actual area being rotated to form the solid.

Integration is the mathematical process that allows us to accumulate the volume of infinitely many small disks across a region. It's an essential tool, enabling us to calculate the exact volume of the solid generated.
For our example, two integrals need evaluation. The integral for \(V_1\) focuses on the region below \(y = x^2\), computed as \( \pi \int_{-2}^{0} x^4 dx \). After integration, we find \(V_1 = \frac{32\pi}{5}\).
The second integral, for \(V_2\), regards the portion under \(y = 2-x\), yielding \(V_2 = \frac{5\pi}{3} \). Once solved, these integrals provide partial volumes of the solid, added together to give a total volume in the final step. Integration ensures precision in building from smaller parts to find the complete volume.