In this section, we will focus on using definite integration to determine the volumes of solids of revolution. These solids are created by revolving a region bounded by curves around an axis (x-axis or y-axis). We'll particularly explore methods to calculate the volume of solids formed by revolving regions between curves, like the volume of a solid formed when revolving the region between $y = 9 - x^2$ and $y = 5$ around the x-axis.

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**Volume of Revolution Between Two Curves**

**With Respect to x**

**Concept**:

Finding the volume of revolution between two curves involves calculating the volumes formed by each curve around the x-axis and subtracting one from the other.

**Procedure**:

a. Ensure $y$ is the subject in the equations of the curves.

b. Apply the formula:

$\pi \int_a^b (y_1^2 - y_2^2) \, \mathrm{dx}$$\int_a^b \pi y_1^2 \, \mathrm{dx} - \int_a^b \pi y_2^2 \, \mathrm{dx}$**With Respect to y**

**Procedure**:

a. Make $x$ the subject in the equations of the curves.

b. Use the formula:

$\pi \int_a^b (x_1^2 - x_2^2) \, \mathrm{dy}$$\int_a^b \pi x_1^2 \, \mathrm{dy} - \int_a^b \pi x_2^2 \, \mathrm{dy}$**Example Questions**

**Problem 1 **:

**Given: **Curve $y = 2(3x - 1)^{\frac{1}{3}}$and lines $x = \frac{2}{3}$, $x = 3$.

**Task: **Calculate the volume when the shaded region is rotated 360° about the x-axis.

**Solution**:

**1. Formula: **Volume of revolution formula:

**2. Integration**:

simplifies to

$\int{\frac{2}{3}}^3 \pi \left( 4(3x - 1)^{\frac{2}{3}} \right) \, \mathrm{dx}$**3. Final Calculation**:

**Problem 2**:

**Given:** The region between the curves $y = x^2$ and $y = x$ in the first quadrant.

**Task:** Find the volume of the solid formed when this region is revolved about the y-axis.

**Solution**:

**1.** **Preparation**: Express $x$ as the subject. Here, $x = \sqrt{y}$ and $x = y$.

**2. Formula**: Apply the formula for revolution about the y-axis:

**3. Integration**:

simplifies to

$\pi \int_0^1 (y^2 - y) \, \mathrm{dy}$**4. Final Calculation**:

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.