MATH 144: Calculus for the Physical Sciences I

Concept 5.3.1. Definite integrals.

Let $f(x)$ be a function defined for $x \in [a,b]\text{.}$ We divide the interval $[a,b]$ into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}\text{.}$ We let $x_0=a\text{,}$ $x_1 = a + \Delta x, \ldots, x_n = b$ be the right endpoints of these intervals. The Riemann sum $R_n$ is defined by

The definite integral of $f$ from $a$ to $b$, denoted by $\int_a^b f(x)\ dx\text{,}$ is the $n\to \infty$ limit of $R_n\text{:}$

provided that the limit exists. If it exists, we say that $f$ is integrable on $[a,b]\text{.}$

In the definition above we used the right-point rule to write down the Riemann sum $R_n\text{.}$ But in fact we can use any point $x_i^* \in [x_{i-1},x_i]$ in the subintervals to define the Riemann sum:

The definite integral is still obtained as the $n\to \infty$ limit of $S_n\text{,}$ and it is equal to the definition above, regardless of the choice of $x_i^*\text{.}$

In the notation $\int_a^b f(x)\ dx\text{,}$ $f(x)$ is called the integrand, and $a$ and $b$ are called the limits of integration: $a$ is the lower limit while $b$ is the upper limit.

Note: the definite integral $\int_a^b f(x)\ dx$ is a number; it is not a function of $x\text{.}$

Concept 5.3.2. Integrable functions.

Many functions are integrable. More precisely, if $f$ is continuous on $[a,b]\text{,}$ or if $f$ has only a finite number of jump discontinuities on $[a,b]\text{,}$ then it is integrable on $[a,b]\text{.}$

Concept 5.3.3. Definite integrals and areas.

If $f(x) \geq 0$ on $[a,b]\text{,}$ then $\int_a^b f(x)\ dx$ calculates the area bounded by $y=f(x)\text{,}$ $y=0\text{,}$ $x=a$ and $x=b\text{.}$

If $f(x) \leq 0$ on $[a,b]\text{,}$ then $\int_a^b f(x)\ dx$ calculates minus the area bounded by $y=f(x)\text{,}$ $y=0\text{,}$ $x=a$ and $x=b\text{.}$

In general, if $f(x)$ is partly positive and partly negative over $[a,b]\text{,}$ then $\int_a^b f(x)\ dx$ calculates the net area, which is the area above the $x$-axis minus the area below the $x$-axis.

Accordingly, the true area (as opposed to the net area) between $y=f(x)\text{,}$ $y=0\text{,}$ $x=a$ and $x=b$ is given by

Concept 5.3.4. Properties of definite integrals.

Many properties of definite integrals can be proved from the geometric interpretation:

1. $\displaystyle \int_a^b f(x)\ dx = - \int_b^a f(x) \ dx\text{,}$
2. $\displaystyle\int_a^a f(x)\ dx = 0\text{,}$
3. $\displaystyle\int_a^b c\ f(x)\ dx = c \int_a^b f(x)\ dx\text{,}$
4. $\displaystyle\int_a^b \left( f(x) \pm g(x) \right)\ dx = \int_a^b f(x)\ dx \pm \int_a^b g(x)\ dx\text{,}$
5. $\displaystyle \int_a^b dx = b-a\text{,}$
6. $\displaystyle\int_a^b f(x)\ dx = \int_a^c f(x)\ dx + \int_c^b f(x)\ dx\text{,}$
7. If $f(x)$ is even, then $\displaystyle \int_{-a}^a f(x)\ dx = 2 \int_0^a f(x)\ dx\text{,}$
8. If $f(x)$ is odd, then $\displaystyle \int_{-a}^a f(x)\ dx = 0\text{,}$
9. If $f(x) \geq 0$ for $x\in [a,b]\text{,}$ then $\displaystyle\int_a^b f(x)\ dx \geq 0\text{,}$
10. If $f(x) \geq g(x)$ for $x \in [a,b]\text{,}$ then $\displaystyle\int_a^b f(x)\ dx \geq \int_a^b g(x)\ dx\text{,}$
11. If $m \leq f(x) \leq M$ for $x \in [a,b]\text{,}$ then
    $$\begin{equation*} m (b-a) \leq \int_a^b f(x)\ dx \leq M(b-a), \end{equation*}$$
12. $$\begin{equation*} \left| \int_a^b f(x) \ dx \right| \leq \int_a^b \left| f(x) \right|\ dx. \end{equation*}$$