15. Computing and Applying Basic Definite Integrals

The key to applying definite integrals lies in modeling. First, clarify what is being accumulated—area, distance, probability, or total cost—then write the corresponding function and interval.

I always assign units to integral results. If the units don’t make sense, it usually means the function or interval was set up incorrectly.

In the previous article, we explored the relationship between definite integrals and area, learning that definite integrals can compute the area under a curve. In this article, we delve deeper into the computation and applications of basic definite integrals to better understand their significance in real-world problems.

1. Definition of the Basic Definite Integral

A definite integral computes the cumulative quantity of a function over a given interval. For a continuous function $f(x)$ defined on the closed interval $[a, b]$, its definite integral is defined as:

When computing a definite integral: first find an antiderivative; then substitute the upper and lower limits; next check the sign of the function over the interval; finally interpret the result as area, accumulated quantity, or average value.

$$\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$

where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is any point in the $i$-th subinterval.

2. Computing Basic Definite Integrals

In this section, we compute several basic definite integrals through illustrative examples.

Before reading “Definite Integrals and Applications: Basic Computation and Applications”, preview the visual path from problem to result shown in this diagram. After reading, revisit the main text to verify whether you can reproduce each step independently.

Example 1: Compute $\int_0^1 x^2 \, dx$

Compute the definite integral of $f(x) = x^2$ over $[0, 1]$:

$$\int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}$$

The result is $\frac{1}{3}$, representing the area between the curve $y = x^2$ and the $x$-axis over $[0, 1]$.

Example 2: Compute $\int_1^2 (3x^2 - 4) \, dx$

Compute the definite integral of $f(x) = 3x^2 - 4$ over $[1, 2]$:

$$\int_1^2 (3x^2 - 4) \, dx = \left[ x^3 - 4x \right]_1^2$$

Evaluating:

$$= (2^3 - 4 \cdot 2) - (1^3 - 4 \cdot 1) = (8 - 8) - (1 - 4) = 0 + 3 = 3$$

The result is $3$, representing the net signed area between $y = 3x^2 - 4$ and the $x$-axis over $[1, 2]$.

3. Applications of Definite Integrals

Definite integrals are not merely tools for computing area—they appear widely across physics, engineering, and other disciplines. Examples include:

Example 3: Computing Displacement Over Time

Suppose an object’s velocity is modeled by $v(t) = 4t^2$. To find its displacement over the time interval $[0, 3]$, integrate velocity:

$$\text{Displacement} = \int_0^3 v(t) \, dt = \int_0^3 4t^2 \, dt$$

Computing:

$$= \left[ \frac{4t^3}{3} \right]_0^3 = \frac{4 \cdot 3^3}{3} - 0 = \frac{4 \cdot 27}{3} = 36$$

Thus, the object’s displacement over $[0, 3]$ is $36$ units.

Example 4: Computing Area of an Irregular Region

We can also use definite integrals to compute areas of irregular regions. For instance, find the area bounded by $y = \sqrt{x}$ and $y = 0$ over $[0, 1]$:

$$\text{Area} = \int_0^1 \sqrt{x} \, dx$$

Computing:

$$= \left[ \frac{2}{3} x^{3/2} \right]_0^1 = \frac{2}{3}(1 - 0) = \frac{2}{3}$$

This region has area $\frac{2}{3}$.

If you haven’t fully internalized “Definite Integrals and Applications: Basic Computation and Applications”, revisit this card and walk through its four steps again.

When reviewing “Definite Integrals and Applications: Basic Computation and Applications”, avoid tackling large projects at once. Instead, start with one simple example to confirm whether the core logic is clear.

4. Summary

In this article, we learned how to compute basic definite integrals and demonstrated their practical applications through concrete examples. Definite integrals serve not only to calculate geometric area but also to model physical quantities—such as displacement—and solve problems across science and engineering.

In the next article, we will introduce multivariable calculus, focusing on multivariable functions and partial derivatives. We encourage you to continue following along and deepen your understanding of this essential topic.