It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.

# Calculus

## The Definite Integral

As discussed in the previous section, the process of Riemann sums leads to the definition of the definite integral. We can denote the definite integral as

$\int_{a}^{b} f(x)dx = \lim_{n\to\infty} \sum\limits_{i=0}^{n-1} f(x_i)\Delta x$

To evaluate a definite integral, we can express it as the difference of two antiderivatives at their end points.

$\int_{a}^{b} f(x)dx = F(b) - F(a)$

where $$F'(x)=f(x)$$.

Example: Evaluate the definite integral $\int_{1}^{4} \frac{y+4}{\sqrt{y}}dy$

Solution:

First, we want to evaluate the integral

$\int \frac{y+4}{\sqrt{y}}dy$

This integral can be simplified by splitting the fraction and simplifying roots into fractional exponents

\begin{align} &\int \frac{y+4}{\sqrt{y}}dy \\=&\int \left(\frac{y}{\sqrt{y}} + \frac{4}{\sqrt{y}}\right)dy \\ =&\int \left( y^{\frac{1}{2}} + 4y^{-\frac{1}{2}} \right)dy \end{align}

Now we can evaluate the integral with constant c

\begin{align} =& \frac{y^{\frac{3}{2}}}{\frac{3}{2}} + \frac{4y^{\frac{1}{2}}}{\frac{1}{2}} + c \\ =& \frac{2y^{\frac{3}{2}}}{3} + 8y^{\frac{1}{2}} + c \end{align}

To evaluate the definite integral, we sub in the end points $$[1,4]$$

\begin{align} F(4) - F(1) =& \left[\frac{2y^{\frac{3}{2}}}{3} + 8y^{\frac{1}{2}} + c \right]_{1}^{4} \\ =& \left[\frac{2(\textbf{4})^{\frac{3}{2}}}{3} + 8(\textbf{4})^{\frac{1}{2}} + c \right] - \left[\frac{2(\textbf{1})^{\frac{3}{2}}}{3} + 8(\textbf{1})^{\frac{1}{2}} + c \right] \\ =& \left[ \frac{64}{3}\right] - \left[ \frac{26}{3} \right] \\ =& \frac{38}{3} \end{align}