Areas Involving Exponential & Logarithmic Functions
Learn areas involving exponential and logarithmic functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). The curves \(y = e^x\) and \(y = \ln x\) are mirror images about the line \(y = x\), so an awkward area beside one curve can often be found using the other.
You will learn to find areas under exponential curves with definite integrals, find areas beside logarithmic curves by integrating with respect to \(y\), and use the inverse function relationship to swap one region for an easier one — a clever, exam-ready strategy.
Theory
Areas under exponential and log curves use the same definite-integral method with the exponential and logarithmic rules. This Year 12 Mathematics Advanced topic (MAV-12-05) evaluates such areas.
Area \(=\displaystyle\int_a^b f(x)\,dx\). Use \(\int e^{ax}dx=\dfrac1a e^{ax}+C\) and \(\int\dfrac1x dx=\ln|x|+C\). \(e^{x}\) is always positive, so its area is just the integral.
\(y=e^{x}\) and \(y=\ln x\) are reflections of each other in the line \(y=x\), so the area between \(y=\ln x\) and the \(y\)-axis from \(y=c\) to \(y=d\) is \(\displaystyle\int_c^d e^{y}\,dy\).
Note: areas enclosed between two curves are studied in Mathematics Extension 1, not in this course.
Method
- Choose the exponential or logarithmic rule.
- Evaluate the primitive at the limits.
- Leave exact answers such as \(e-1\).
| \(\ \) | \(=\) | \(\big[e^{x}\big]_0^1=e-1\) |
| \(\ \) | \(=\) | \(\big[\tfrac12 e^{2x}\big]_0^1=\dfrac{e^{2}-1}{2}\) |
| \(\ \) | \(=\) | \(\big[\ln x\big]_1^{e}=1\) |
| \(\ \) | \(=\) | \(\big[e^{x}\big]_0^2=e^{2}-1\) |
Common pitfalls
Frequently asked questions
What is the area under e to the x from 0 to 1?
It is e minus 1, from e to the x evaluated between 0 and 1.
How do you find the area under e to the 2x?
Integrate to get one half e to the 2x, then evaluate between the limits.
What is the area under 1 over x from 1 to e?
It is ln e minus ln 1, which is 1 minus 0, so 1.
Should you round these areas?
Leave exact answers like e minus 1 unless the question asks for a decimal approximation.