Applications of Integration — Study Notes
Applications of Integration — Study Notes
1. The definite integral as a limit of a sum
Partition \([a,b]\) into \(n\) sub-intervals and form a Riemann sum \(\sum_{i} f(t_i)\,\Delta x_i\). As the partition is refined (each \(\Delta x_i\to 0\)), the sum approaches a single number, written \(\displaystyle\int_{a}^{b} f(x)\,dx\). For a positive continuous function this number equals the geometric area under the curve.
Two convenient closed forms obtained by taking equal sub-intervals are
\[ \int_{a}^{b} f(x)\,dx = \lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^{n} f\!\left(a+i\,\frac{b-a}{n}\right). \]2. The two Fundamental Theorems
First theorem: if \(F(x)=\displaystyle\int_{a}^{x} f(u)\,du\), then \(F'(x)=f(x)\). In words, differentiating an integral with respect to its upper limit returns the integrand — this is exactly what shortcuts questions such as "find \(\tfrac{df}{dx}\)".
Second theorem: if \(F\) is any antiderivative of \(f\), then \(\displaystyle\int_{a}^{b} f(x)\,dx = F(b)-F(a)\). The arbitrary constant cancels, so a definite integral has a unique value.
3. Properties used to simplify definite integrals
These are the workhorses of the chapter and appear constantly in the MCQs:
- Reflection: \(\displaystyle\int_{0}^{a} f(x)\,dx=\int_{0}^{a} f(a-x)\,dx\). The "king's rule" version \(\int_{a}^{b} f = \int_{a}^{b} f(a+b-x)\) lets you add the integral to a reflected copy.
- Even/odd symmetry: on \([-a,a]\), \(\int = 2\int_{0}^{a} f\) if \(f\) is even, and \(\int = 0\) if \(f\) is odd.
- Period splitting: \(\int_{0}^{2a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx\) when \(f(2a-x)=f(x)\), and \(=0\) when \(f(2a-x)=-f(x)\).
4. Reduction (Wallis) formulae
For non-negative integers the standard values are extremely useful for powers of sine and cosine:
\[ \int_{0}^{\pi/2}\sin^{n}x\,dx=\int_{0}^{\pi/2}\cos^{n}x\,dx = \begin{cases}\dfrac{(n-1)(n-3)\cdots 1}{n(n-2)\cdots 2}\cdot\dfrac{\pi}{2}, & n \text{ even},\\[2mm] \dfrac{(n-1)(n-3)\cdots 2}{n(n-2)\cdots 3}, & n \text{ odd}.\end{cases} \]For odd \(n\) there is no factor of \(\pi/2\); for even \(n\) there is. Mixing this up is a frequent error.
5. Improper integrals and the gamma integral
Integrals over an infinite range, such as \(\int_{0}^{\infty}\), are evaluated as limits. The key result is the gamma integral
\[ \int_{0}^{\infty} e^{-ax}\,x^{n}\,dx = \frac{n!}{a^{\,n+1}}, \qquad \Gamma(n)=\int_{0}^{\infty}e^{-x}x^{n-1}\,dx=(n-1)!, \]together with the recurrence \(\Gamma(n+1)=n\,\Gamma(n)\), which turns gamma equations into simple algebra.
6. Area of a plane region
Area is a geometric (non-negative) quantity, so signs matter:
- Above the x-axis between \(x=a\) and \(x=b\): \(A=\displaystyle\int_{a}^{b} y\,dx\).
- Below the x-axis: \(A=\Big|\displaystyle\int_{a}^{b} y\,dx\Big|\) (take the magnitude).
- To the right/left of the y-axis: integrate \(x\,dy\) similarly.
- Between two curves with \(f\ge g\): \(A=\displaystyle\int_{a}^{b}\big(f(x)-g(x)\big)\,dx\) (upper minus lower); for left–right boundaries use \(\int_{c}^{d}\big(x_R-x_L\big)\,dy\).
Common trap: when a curve crosses the axis (e.g. \(\sin x\) on \([0,2\pi]\)), split at the crossing and add absolute values, otherwise the positive and negative parts cancel and you under-report the area.
7. Volume of a solid of revolution
Rotating a region about an axis sweeps out a solid whose cross-sections are discs:
\[ \text{about the x-axis: } V=\pi\int_{a}^{b} y^{2}\,dx, \qquad \text{about the y-axis: } V=\pi\int_{c}^{d} x^{2}\,dy. \]Always rewrite the boundary in the right variable before integrating (e.g. express \(x\) in terms of \(y\) for y-axis rotation) and read the limits off the axis of rotation.
Quick exam reminders
- Test even/odd before integrating over a symmetric interval — it often kills the whole integral or halves the work.
- For \(\int_{0}^{\pi}\dfrac{dx}{1+c^{\cos x}}\)-type problems, add the reflected copy; the answer is half the interval length.
- Distinguish "value of the integral" (signed) from "area" (always positive).
- For odd powers in a Wallis integral, no \(\pi/2\) factor appears.