Area Calculation with Integrals

We know from using the Newton-Leibniz theorem that the area under the curve of a function $f(x)$ with primitive $F(x)$ taken from $x=a$ to $x=b$ is given by:

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

The reality is that we need to adapt this depending on the function. This formula represents the net area between the curve of a function $f(x)$ and the $x$ -axis.

Table of Contents

Motivation

Say we want to find the total area enclosed between the graph of $f(x) = x^3$ and the $x$ -axis from $x = -1$ to $x = 1$ .

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If we blindly compute the definite integral across the entire interval using the Newton-Leibniz formula, we get:

$\int_{-1}^{1} x^3 \, dx = \frac{x^4}{4} \Bigg|_{-1}^{1} = \frac{1}{4} - \frac{(-1)^4}{4} = 0$

However, the total physical area cannot be zero because we see two regions and area is always non-negative.

How do we fix this? The problem comes from our function changing sign within the given interval. Therefore, if we put our function in absolute value, that will fix our problem. However, this is only helpful when we have a calculator to find the integral. We should still learn how to do this without relying on a calculator.

Total Area

To fix our opening problem, we follow the steps below:

Separate the integral into appropriate intervals: Identify where the function changes signs by finding its roots ( $x$ -intercepts).
Set up the area formula with a minus sign for negative regions: To prevent cancellation, we place a minus sign in front of every integral for which the function is below the $x$ -axis in that interval of integration.
Evaluate each integral separately via Newton-Leibniz.
Add the results.

Example: Find the total area enclosed between the graph of $f(x) = x^3$ and the $x$ -axis from $x = -1$ to $x = 1$ .

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The definite integral tracks signed area. On the interval $[-1, 0]$ , the function dips below the $x$ -axis, yielding a negative integral value ( $A_1 = -0.25$ ). On $[0, 1]$ , the function stays above the axis, yielding an equal positive value ( $A_2 = 0.25$ ). When integrated as a single piece, these regions cancel each other out ( $0.25 - 0.25 = 0$ ).

Let us see another example.

Example: Below is the graph of $f(x)=x^3 - 2x^2 - x + 2$ . Find the total geometric area enclosed between $f(x)$ and the $x$ -axis from $x = -1$ to $x = 2$ .

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We can either find $A_1$ and $A_2$ as areas (and so they cannot be negative) and simply say

$Area=A_1+A_2$

Or find $A_1$ and $A_2$ as integrals (so $A_2$ will be negative because the function is below the $x$ -axis) and then subtract them, as in the example above.

We may not always be given the graph of the function, so it may be up to use to understand if it is above the $x$ -axis or below the $x$ -axis in each interval.

Area Between Curves

If we are given two functions $f(x)$ and $g(x)$ and asked for the area between them, it is implied that these functions will intersect. To find the area between the two functions, we could find the area below the graph of a function, the area below the other, and subtract them. However, an easier approach would be to take the difference of functions. This will be a new function and the area between this new function and the $x$ -axis is equivalent to the area between the functions.

The only problem with this is that we need to account for where the new function is positive/negative. This is the same as understand which function $f$ or $g$ is the greater one.

Example: Find the total geometric area enclosed between the cubic polynomial $f(x) = x^3 - x^2$ and the linear function $g(x) = 2x$ .

To see where the curves cross and which function sits on top over different intervals, let us inspect the graph below:

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Exercises

Find the exact geometric area enclosed between the graph of each function and the $x$ -axis over the given interval. Use the provided vector plots for reference.

1. $f(x) = 2x + 1$ from $x = 0$ to $x = 3$ .

2. $f(x) = 4 - x^2$ from $x = 0$ to $x = 2$ .

3. $f(x) = x^3$ from $x = 0$ to $x = 2$ .

4. $f(x) = e^x$ from $x = 0$ to $x = 1$ .

Find the area enclosed between the curves of the given functions. Determine their intersection points algebraically to find the limits of integration.

1. Enclosed by $f(x) = x + 2$ and $g(x) = x^2$ .