2. The volume of a right circular cone is Irh where r is the radius of the base and h is the height of a cone. Use the disk method to derive this formula.
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Volume of a Right Circular Cone**
**Problem Statement:**
The volume of a right circular cone is \(\frac{\pi r^2 h}{3}\) where \(r\) is the radius of the base and \(h\) is the height of the cone. Use the disk method to derive this formula.
**Detailed Solution:**
**Step-by-step Derivation Using Disk Method:**
1. **Understanding the Geometry:**
- Consider a right circular cone with height \(h\) and base radius \(r\).
- The cone is oriented such that its vertex is at the origin (0,0).
2. **Disk Method Concept:**
- Imagine slicing the cone into very thin horizontal disks (circular slices) perpendicular to the \(y\)-axis.
- We'll integrate these disks' volumes from the base to the apex of the cone.
3. **Equation of the Cone's Surface:**
- At a height \(y\), the radius of the disk is proportional to \(y\), given by \(r(y) = \frac{r}{h}y\).
4. **Volume of a Thin Disk (dV):**
- Each disk has a small thickness \(dy\) and radius \(r(y)\).
- Volume of each disk: \(dV = \pi \left(\frac{r}{h} y \right)^2 dy\).
5. **Integrating to Find the Cone's Total Volume:**
- Integrate \(dV\) from the base (\(y=0\)) to the top (\(y=h\)):
\[
V = \int_{0}^{h} \pi \left(\frac{r}{h} y \right)^2 dy
\]
6. **Simplifying the Integral:**
- \(V = \pi \int_{0}^{h} \left(\frac{r^2}{h^2} y^2 \right) dy\)
- \(V = \pi \frac{r^2}{h^2} \int_{0}^{h} y^2 dy\)
7. **Evaluating the Definite Integral:**
- The integral of \(y^2\) gives: \(\int_{0}^{h} y^2 dy = \left[ \frac{y^3}{3} \right]_{0}^{h}\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F777e0514-8160-4940-b57c-96884b4e2915%2F8233e588-3b81-4d60-a320-261d85fe45a2%2F6hkk4hl_processed.png&w=3840&q=75)
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