Area and Definite Integrals

Section 4.3 Area and Definite Integrals

Definite integrals give numbers as answers and the number refers to an amount of area. (Previously they were functions.)

Consider the following:

We often start by approximating this area by the area of related rectangles.
Let's use 7 rectangles. Also we will make them left-handed.

The area under the curve from the previous graph, take \(f\) to be a non-negative function on \([a,b]\) and let \(R\) be the region bounded above by the function values, below by the x-axis, and left and right by \(x = a\) and \(x = b\text{.}\)

We can:

Divide the internal \([a,b]\) into \(n\) equally-sized subintervals by spacing \(n - 1\) points evenly between \(a\) and \(b\) along the x-axis. (Call these points: \(x\text{,}\) \(x_1\text{,}\) \(x_2\text{,}\) ... \(x_{n-1};\) each interval has a width of \(\Delta x = \frac{b-a}{n}\))
We let \(x_i^*\) be the point in interval \(i\) where we measure the height of the function (consider left, right, or center). The areas of rectangles are: \(f(x_1^*) \Delta x\text{,}\) \(f(x_2^*) \Delta x\text{,}\) ... , \(f(x_n^*) \Delta x\)
The area is approximated as:

\begin{equation*} A = area(R) \approx f(x_1^*) \Delta x + f(x_2^*) \Delta x + ... + f(x_n^*) \Delta x \end{equation*}

\begin{equation*} \text{ or } A \approx \sum \limits_{k=1}^{n}f(x_k^*) \Delta x \end{equation*}
We can do better with using more and more rectangles to be more exact:

\begin{equation*} A = \lim_{n \to \infty} \sum \limits_{k=1}^{n}f(x_k^*) \Delta x \end{equation*}

We definte this as something called the definite or Rieman integral. Rieman integral =

\begin{equation*} \int_a^b f(x) dx = \lim_{max(\Delta x) \to 0} \sum \limits_{k=1}^{n}f(x_k^*) \Delta x \end{equation*}

Theorem 4.3.2.

If \(f\) is continuous on \([a,b]\) and the net signed area, \(A\) between the graph of \(f\) on the interval \([a,b]\) is \(A = \displaystyle \int_a^b f(x) dx\text{.}\)

Example 4.3.3.

Figure 4.3.4.

what is the area under the curve from \(x = 0\) to \(x = 1\text{?}\)

\begin{equation*} A = \int_0^1x dx = \frac{1}{2} \end{equation*}
what is the net signed are from \(x = -1\) to \(x = 0\text{?}\)

\begin{equation*} A = \int_0^1 x dx = - \frac{1}{2} \end{equation*}
what is the net signed area from \(x = -1\) to \(x = 1\text{?}\)

\begin{equation*} A = \int_{-1}^1 x dx = 0 \end{equation*}

Definition 4.3.5.

Let \(a \lt b\text{,}\) \(f\) and \(g\) are integrable, \(c\) is a constant:

\begin{align*} \int_a^b f(x) dx \amp = 0 \\ \int_a^b f(x) dx \amp = -\int_a^b f(x) dx \\ \int_a^b c \cdot f(x) dx \amp = c \cdot \int_a^b f(x) dx \\ \int_a^b f(x) \pm g(x) dx \amp = \left( \int_a^b f(x) dx \right) \pm \left( \int_a^b g(x) dx \right) \end{align*}

Example 4.3.6.

Find \(\displaystyle \int_0^1 3x dx\text{.}\)

Solution

\begin{align*} \int_0^1 3x dx \amp = 3 \cdot \int_0^1 x dx\\ \amp = 3 (1\frac{1}{2})\\ \amp = \frac{3}{2} \end{align*}