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*}