91Ó°ÊÓ

The Disk Method is a way to find the volume of a solid of revolution. When you rotate a region around an axis, you get a 3-dimensional shape. Imagine slicing this shape perpendicular to the axis of rotation. Each slice is like a disk or a circle. The volume of the solid can be found by summing up the volumes of these infinitesimally thin disks. The formula for the volume using the disk method is:

\[ V = \int_{a}^{b} \pi r^{2} dy \]

where \pi r^{2} is the area of each disk, and \int_{a}^{b} represents the sum of these areas from an interval \[a, b\]. The radius \((r)\) of each disk will depend on the function you are rotating.

In our problem, we rotate the region around the \((y)\)-axis. So, the radius \((r)\) is found by solving the given equation \((y=x^6)\) for \((x)\), which gives \((x=y^{1/6})\). Hence, the radius \((r)\) is \((y^{1/6})\) for our integral.

Integration is a fundamental concept in calculus that helps calculate volumes, areas, and other quantities. In this exercise, we use integration to sum up the areas of infinitely many disks to find the volume of the solid. The integral we need to solve is:

\[ V = \int_{0}^{1} \pi y^{1/3} dy \]

Integrating the function \((y^{1/3})\) means finding the antiderivative, which is a function whose derivative is \((y^{1/3})\). To integrate \((y^{1/3})\), follow these steps:

- Increase the exponent by 1:
So, from \((y^{1/3})\), the new exponent is \((1/3 + 1 = 4/3)\).

- Divide by the new exponent:
The antiderivative becomes \((y^{4/3} \/ (4/3))\).

- Simplify:
This further simplifies to \((3/4 \cdot y^{4/3})\).

Finally, apply the limits of integration \((0 \text{ and } 1)\) to the antiderivative found:

\[ V = \pi \[ \frac{3}{4} y^{4/3} \]_{0}^{1} = \pi \left( \frac{3}{4} \cdot (1)^{4/3} - \frac{3}{4} \cdot (0)^{4/3} \right) \]

Solving this, we get the volume \((V)\) as \((\frac{3 \pi}{4})\).

A definite integral represents the net area under a curve between two points. It is one of the key tools for calculating exact values, such as areas, volumes, and other quantities. In this problem, the definite integral helps us find the exact volume of the solid formed by rotating the given region.

When setting up a definite integral, you need:

- Function to integrate: In our problem, it’s \((y^{1/3})\).
- Limits of integration: These are the bounds \((0 \text{ to } 1)\) for our \((y)\) values.

The process of solving a definite integral involves:

- Finding the antiderivative of the function.
- Applying the Fundamental Theorem of Calculus, which states that to evaluate the definite integral \((\int_{a}^{b} f(y) dy)\), you subtract the value of the antiderivative at the lower limit \((a)\) from its value at the upper limit \((b)\).

In our problem, the definite integral calculation is:

\[ V = \pi \left[ \frac{3}{4} y^{4/3} \right]_{0}^{1} \]

This represents the exact volume of the solid when evaluating from \((0)\) to \((1)\)