# Volumes using Integration

## Volume of Revolution

We can use integration to find volumes of revolution between $x=a$ and $x=b$.
When the region enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$ is revolved through $2 \pi$ or $360^{\circ}$about the $x$-axis to generate a solid, the volume of the solid is given by:
\begin{align} \displaystyle V &= \lim_{h \rightarrow 0} \sum_{x=a}^{x=b}{\pi \big[f(x)\big]^2 h} \\ &= \int_{a}^{b}{\pi \big[f(x)\big]^2}dx \\ &= \pi \int_{a}^{b}{y^2}dx \end{align}

### Example 1

Find the volume of the solid generated when the line $y=x$ for $1 \le x \le 3$ is revolved through $2 \pi$ or $360^{\circ}$ around the $x$-axis.

\begin{align} \displaystyle V &= \pi \int_{1}^{3}{y^2}dx \\ &= \pi \int_{1}^{3}{x^2}dx \\ &= \pi \Big[\dfrac{x^3}{3}\Big]_{1}^{3} \\ &= \dfrac{\pi}{3} \big[x^3\big]_{1}^{3} \\ &= \dfrac{\pi}{3} \big[3^3-1^3\big] \\ &= \dfrac{\pi}{3} \times 26 \\ &= \dfrac{26 \pi}{3} \text{ units}^3 \end{align}

### Example 2

Find the volume of the solid generated when the line $y=\sqrt{x}$ for $0 \le x \le 2$ is revolved through $2 \pi$ or $360^{\circ}$ around the $x$-axis.

\begin{align} \displaystyle V &= \pi \int_{0}^{2}{y^2}dx \\ &= \pi \int_{0}^{2}{\sqrt{x}^2}dx \\ &= \pi \int_{0}^{2}{x}dx \\ &= \pi \Big[\dfrac{x^2}{2}\Big]_{0}^{2} \\ &= \dfrac{\pi}{2}\pi \big[x^2\big]_{0}^{2} \\ &= \dfrac{\pi}{2}\pi \big[2^2-0^2\big]_{0}^{2} \\ &= \dfrac{\pi}{2} \times 4 \\ &= 2 \pi \text{ units}^3 \end{align} ## Simplified Calculus: Exploring Differentiation by First Principles

Exploring differentiation by first principles Suppose we are given a function $f(x)$ and asked to find its derivative at the point where $x=a$. This is…

## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…

## Trigonometry Made Easy: Integration by Parts Demystified

Integration by Parts is made of the product rule of differentiation. The derivative of $uv$ is $u’v + uv’$ and integrates both sides. \( \begin{aligned}…

## Sigma Notation | Summation Notation | Sum of an Arithmetic Series

Another mathematical device widely used in sequences and series is $\textit{sigma notation}$. The Greek letter $\sum$ (capital sigma), indicates the sum of a sequence. For…

## Calculation of Areas under Curves

Consider the function $f(x)=x^2+2$.We wish to estimate the green area enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=4$. Suppose we divide…