Areas Involving Exponential & Logarithmic Functions
Areas involving exponential and logarithmic functions use definite integrals of \(e^{x}\), \(\ln x\) and related curves to find the regions they bound.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Integral calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.
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.