Review 2: Definite Integrals

4.1 Definition

Definite integral is defined in terms of Riemann sums (we are skipping them here) and can be interpreted as "area under the graph of $y=f(x)$" when $f(x)\ge 0$ (graph of $f(x)$ is above the x-axis).

4.2 Notation

We commonly use the following notation for definite integrals:

\(\int_a^b\; f(x)\; dx\\)

where a and b are endpoints of an interval.

4.3 The Fundamental Theorem of Calculus

If a function is continuous on an interval $[a,b]$ (imagine you can draw its graph over this interval without ever lifting your pen off the paper), then there always exists an antiderivative$F(x)$ of $f$, and we have

$\int_a^b\; f(x)\; dx\; =\; F(b)-F(a)$

4.4 Example

1. $\int_0^1 3x^5\; dx\; =\; \frac{1}{2}x^6\Big |_0^1=\frac{1}{2}(1)^6-\frac{1}{2}(0)^6=\frac{1}{2}$. I could have also chosen $\frac{1}{2}x^6 +47$ as an antiderivative for these computations.

This is also a valid solution:

$\int_0^1 3x^5\; dx\; =\;\left( \frac{1}{2}x^6+47\right)\Big |_0^1=\left(\frac{1}{2}(1)^6+47\right)-\left(\frac{1}{2}(0)^6+47\right)=\frac{1}{2}$

as well as, 

$\int_0^1 3x^5\; dx\; =\;\left( \frac{1}{2}x^6+C\right)\Big |_0^1=\left(\frac{1}{2}(1)^6+C\right)-\left(\frac{1}{2}(0)^6+C\right)=\frac{1}{2}.$

For ease of computation, we usually use the antiderivative with no C (or C = 0).

2. $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \sin(2x)\; dx\; =\; \frac{-1}{2}\cos(2x)\Big |_{\frac{\pi}{3}}^{\frac{\pi}{2}}= \frac{-1}{2}\cos(\frac{2\pi}{2})-\frac{-1}{2}\cos(\frac{2\pi}{3})$