# Solids of Revolutions Using Definite Integrals

### Solids of Revolutions Using Definite Integrals:

1. Objective:

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

• The method involves revolving a function $y=f\left(x\right)$ around the x-axis or y-axis to form a solid.
• Disk method: Rotating about the x-axis.
• Washer method: Rotating between two functions about the x-axis.
• Shell method: Rotating about the y-axis.
3. Disk Method (Rotating Around x-axis):

• Formula: The volume of the solid formed by rotating the curve $y=f\left(x\right)$ over $\left[a,b\right]$ around the x-axis is given by: $V=\pi {\int }_{a}^{b}\left[f\left(x\right){\right]}^{2}\text{\hspace{0.17em}}dx$
4. Washer Method (Rotating Between Curves Around x-axis):

• Formula: The volume of the solid between two curves $y=f\left(x\right)$ and $y=g\left(x\right)$ over $\left[a,b\right]$ rotated around the x-axis is given by: $V=\pi {\int }_{a}^{b}\left[{\left(f\left(x\right)\right)}^{2}-{\left(g\left(x\right)\right)}^{2}\right]\text{\hspace{0.17em}}dx$
5. Shell Method (Rotating Around y-axis):

• Formula: The volume of the solid formed by rotating the curve $x=f\left(y\right)$ over $\left[c,d\right]$ around the y-axis is given by: $V=2\pi {\int }_{c}^{d}x\cdot f\left(y\right)\text{\hspace{0.17em}}dy$
6. Steps to Find Volume:

• Step 1: Identify Rotation Axis: Determine whether the rotation is around the x-axis or y-axis.

• 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.

7. 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 $\left[0,2\right]$. Let's find the volume of the solid formed by rotating this curve around the x-axis.

1. Objective:

• Calculate the volume of the solid generated by revolving $y={x}^{2}$ over $\left[0,2\right]$ around the x-axis using the disk method.
2. Disk Method Formula:

• The formula for volume using the disk method is: $V=\pi {\int }_{a}^{b}\left[f\left(x\right){\right]}^{2}\text{\hspace{0.17em}}dx$
3. Steps to Find Volume:

• Step 1: Identify Rotation Axis: The rotation is around the x-axis.

• Step 2: Setup Integral for Disk Method:

• The formula for the disk method: $V=\pi {\int }_{a}^{b}\left[f\left(x\right){\right]}^{2}\text{\hspace{0.17em}}dx$.
• For $y={x}^{2}$ over $\left[0,2\right]$, the integral becomes: $V=\pi {\int }_{0}^{2}\left({x}^{2}{\right)}^{2}\text{\hspace{0.17em}}dx=\pi {\int }_{0}^{2}{x}^{4}\text{\hspace{0.17em}}dx$
• Step 3: Evaluate Integral:

• Compute the definite integral: $V=\pi \cdot {\left[\frac{{x}^{5}}{5}\right]}_{0}^{2}=\pi \cdot \left(\frac{{2}^{5}}{5}-\frac{{0}^{5}}{5}\right)=\pi \cdot \frac{32}{5}=\frac{32\pi }{5}$
4. Result:

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