Solids of Revolutions Using Definite Integrals
Solids of Revolutions Using Definite Integrals:

Objective:
 Calculate volumes of solids formed by rotating a curve around a specific axis using definite integrals.

Approach:
 The method involves revolving a function $y=f(x)$ around the xaxis or yaxis to form a solid.
 Disk method: Rotating about the xaxis.
 Washer method: Rotating between two functions about the xaxis.
 Shell method: Rotating about the yaxis.

Disk Method (Rotating Around xaxis):
 Formula: The volume of the solid formed by rotating the curve $y=f(x)$ over $[a,b]$ around the xaxis is given by: $V=\pi {\int}_{a}^{b}[f(x){]}^{2}\text{\hspace{0.17em}}dx$

Washer Method (Rotating Between Curves Around xaxis):
 Formula: The volume of the solid between two curves $y=f(x)$ and $y=g(x)$ over $[a,b]$ rotated around the xaxis is given by: $V=\pi {\int}_{a}^{b}[{(f(x))}^{2}{(g(x))}^{2}]\text{\hspace{0.17em}}dx$

Shell Method (Rotating Around yaxis):
 Formula: The volume of the solid formed by rotating the curve $x=f(y)$ over $[c,d]$ around the yaxis is given by: $V=2\pi {\int}_{c}^{d}x\cdot f(y)\text{\hspace{0.17em}}dy$

Steps to Find Volume:

Step 1: Identify Rotation Axis: Determine whether the rotation is around the xaxis or yaxis.

Step 2: Set Up Integral: Based on the method (disk, washer, or shell), set up the appropriate definite integral formula to calculate volume.

Step 3: Define Limits: Define the limits of integration ($a$, $b$, $c$, $d$) corresponding to the interval of the curve or region being rotated.

Step 4: Evaluate Integral: Compute the definite integral to find the volume of the solid of revolution.


Applications:

Engineering: Used to calculate volumes of components, such as pipes, tanks, and machine parts.

Physics: Applied to find volumes of rotated shapes in thermodynamics, fluid dynamics, and mechanics.

Example: Disk Method for Solid of Revolution
Consider the curve $y={x}^{2}$ over the interval $[0,2]$. Let's find the volume of the solid formed by rotating this curve around the xaxis.

Objective:
 Calculate the volume of the solid generated by revolving $y={x}^{2}$ over $[0,2]$ around the xaxis using the disk method.

Disk Method Formula:
 The formula for volume using the disk method is: $V=\pi {\int}_{a}^{b}[f(x){]}^{2}\text{\hspace{0.17em}}dx$

Steps to Find Volume:

Step 1: Identify Rotation Axis: The rotation is around the xaxis.

Step 2: Setup Integral for Disk Method:
 The formula for the disk method: $V=\pi {\int}_{a}^{b}[f(x){]}^{2}\text{\hspace{0.17em}}dx$.
 For $y={x}^{2}$ over $[0,2]$, the integral becomes: $V=\pi {\int}_{0}^{2}({x}^{2}{)}^{2}\text{\hspace{0.17em}}dx=\pi {\int}_{0}^{2}{x}^{4}\text{\hspace{0.17em}}dx$

Step 3: Evaluate Integral:
 Compute the definite integral: $V=\pi \cdot {\left[\frac{{x}^{5}}{5}\right]}_{0}^{2}=\pi \cdot (\frac{{2}^{5}}{5}\frac{{0}^{5}}{5})=\pi \cdot \frac{32}{5}=\frac{32\pi}{5}$


Result:
 The volume of the solid generated by rotating the curve $y={x}^{2}$ over $[0,2]$ around the xaxis using the disk method is $\frac{32\pi}{5}$ cubic units.