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.
\( \begin{align} \displaystyle
V &= \pi \int_{0}^{2}{y^2}dx \\
&= \pi \int_{0}^{2}{\sqrt{x}^2}dx \\
&= \pi \int_{0}^{2}{x}dx \\
&= \pi \Big[\dfrac{x^2}{2}\Big]_{0}^{2} \\
&= \dfrac{\pi}{2}\pi \big[x^2\big]_{0}^{2} \\
&= \dfrac{\pi}{2}\pi \big[2^2-0^2\big]_{0}^{2} \\
&= \dfrac{\pi}{2} \times 4 \\
&= 2 \pi \text{ units}^3
\end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses