QCE Specialist Mathematics - Unit 4 - Applications of integral calculus

Areas, Volumes and Simpson's Rule | QCE Specialist Mathematics

Learn QCE Specialist applications of integration, including areas between curves, solids of revolution and Simpson's rule.

Updated 2026-05-18 - 4 min read

Applications of integration ask what a changing quantity accumulates into. In Specialist, the big applications are areas between curves, volumes of revolution and numerical approximation using Simpson's rule.

Area between curves

If the upper curve is $f(x)$ and the lower curve is $g(x)$ on $[a,b]$, then:

$ A=\int_a^b [f(x)-g(x)]\,dx. $

The word upper is not decoration. You must know which function is higher on the interval you are integrating across.

If two curves switch order, split the interval at their intersection point. That prevents signed areas from cancelling.

When the region is easier to describe horizontally, integrate with respect to $y$ instead. Then the area is:

$ A=\int_c^d [x_{\text{right}}(y)-x_{\text{left}}(y)]\,dy. $

The same principle is being used: outside boundary minus inside boundary in the direction you are slicing.

Regions are defined by boundaries

The same pair of curves can create different areas depending on the other boundaries in the question. If a vertical line such as $x=3$ is added, the region may no longer run only between the two intersection points. You must sketch or reason through the actual enclosed region before writing the integral.

For example, suppose $y=x^2$ and $y=2x$ intersect at $x=0$ and $x=2$. The area between the curves on that natural enclosed region is:

$ \int_0^2(2x-x^2)\,dx. $

But if the region is also bounded by $x=3$, then part of the boundary may involve a different "top minus bottom" arrangement or a different interval. The integral is chosen from the region, not just from the curve names.

Compound regions should be split at intersections or turning points where the upper/lower relationship changes. If solving for $x$ in terms of $y$ creates two branches, choose the branch that actually forms the right or left boundary of the shaded region. A quick sketch saves a lot of algebra here.

Volumes of revolution

When a region rotates around an axis, it sweeps out a solid. For rotation around the $x$-axis, the disk formula is:

$ V=\pi\int_a^b y^2\,dx. $

If the region is between two curves, use a washer:

$ V=\pi\int_a^b \left(y_{\text{outer}}^2-y_{\text{inner}}^2\right)\,dx. $

For rotation around the $y$-axis, write the radius in terms of $x$ and integrate with respect to $y$:

$ V=\pi\int_c^d x^2\,dy. $

If there is a gap between the axis and the region, use washers rather than disks. Squaring the radius happens before subtracting:

$ V=\pi\int_a^b \left(R^2-r^2\right)\,dx. $

Do not use $\pi\int(R-r)^2\,dx$; that describes a different radius.

Simpson's rule

Simpson's rule estimates a definite integral by fitting parabolic arcs rather than straight line segments. It requires an even number of subintervals.

If

$ w=\frac{b-a}{n}, $

then:

$ \int_a^b f(x)\,dx \approx \frac{w}{3} \left[ f(x_0)+4(f(x_1)+f(x_3)+\cdots)+2(f(x_2)+f(x_4)+\cdots)+f(x_n) \right]. $

With and without technology

The syllabus expects these applications both with and without technology. Without technology, you should be able to set up and evaluate manageable integrals exactly. With technology, the important skill is often setting up the correct integral, intersection points, bounds or Simpson table before using a calculator.

For Simpson's rule, the width is:

$ w=\frac{b-a}{n}. $

The $x$ values must be evenly spaced. If the table spacing is irregular, Simpson's rule in this form does not apply.

Simpson table setup

For Simpson's rule, an even number of subintervals means an odd number of function values. If $n=4$, the table has:

$ x_0,x_1,x_2,x_3,x_4. $

On $[0,2]$ with $n=4$, the width is:

$ w=\frac{2-0}{4}=0.5. $

For $f(x)=e^x$, Simpson's estimate is:

$ \int_0^2 e^x\,dx \approx \frac{0.5}{3} \left[ e^0+4e^{0.5}+2e^1+4e^{1.5}+e^2 \right]. $

The exact value is:

$ \int_0^2 e^x\,dx=e^2-1. $

Comparing an approximate value with the exact value is a good way to check whether the coefficient pattern and spacing were used correctly. There is one more odd-index interior value than even-index interior value because the pattern begins and ends with a $4$ next to the endpoints.

Quick check

Sources

• QCAA Specialist Mathematics subject page
• QCAA Specialist Mathematics 2025 syllabus
• OpenStax Calculus Volume 2: Determining Volumes by Slicing
• OpenStax Calculus Volume 2: Numerical Integration