Volumes by Cross Sections


Volumes by Cross Sections can be setting a unit volume, \( \delta V \) of a cross section. Then establish an integration for evaluating the volume of the entire solid.

Worked Examples of Volumes by Cross Sections

The base of a solid is formed by the area bounded by \( y = \cos x\) and \( y = -\cos x \) for \( \displaystyle 0 \le x \le \frac{\pi}{2} \). Vertical cross sections of the solid taken parallel to the \(y\)-axis are in the shape of isosceles triangles with the equal sides of length 1 unit. Find the volume of the solid.


\( \begin{aligned} \displaystyle
\text{base of the triangle } &= 2 \cos x \\
\text{height of the triangle } &= \sqrt{1- \cos^2 x} \\
\delta V &= \frac{1}{2} \times 2 \cos x \times \sqrt{1- \cos^2 x} \times \delta x \\
&= \cos x \times \sin x \delta x \\
&= \frac{1}{2} \sin 2x \delta x \\
V &= \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \sin 2x dx \\
&= \frac{1}{2}\Big[-\frac{1}{2} \cos 2x \Big]_{0}^{\frac{\pi}{2}} \\
&= -\frac{1}{4}\Big[\cos 2x \Big]_{0}^{\frac{\pi}{2}} \\
&= -\frac{1}{4}\Big[\cos \pi – \cos 0\Big] \\
&= -\frac{1}{4}\Big[-1-1] \\
&= \frac{1}{2} \\
\end{aligned} \\ \)

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials 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 Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Your email address will not be published. Required fields are marked *