# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/integral/definite-integral
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/integral/definite-integral/en.mdx

Output docs content for large language models.

---

export const metadata = {
    title: "Definite Integral",
    description: "Transform Riemann sums into exact area calculations using definite integrals. Learn limit processes and notation with detailed worked examples.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "05/26/2025",
    subject: "Integrals",
};

## From Approximation to Exact Area

In the [Riemann Sums](/subject/high-school/12/mathematics/integral/riemann-sum) material, we learned how to approximate the area under a curve by dividing it into many rectangles. We also know that the more rectangles we use (the larger the value of <InlineMath math="n" />), the more accurate our area approximation becomes.

Now, imagine if we could divide the area into an **infinite number of rectangles**. The width of each rectangle (<InlineMath math="\Delta x" />) would become infinitesimally small, approaching zero. This process of taking the limit as the number of partitions <InlineMath math="n" /> approaches infinity is what transforms the Riemann Sum from a mere approximation into an exact calculation. This very concept gives rise to the **Definite Integral**.

## Notation and Meaning of the Definite Integral

The definite integral is the formal way of writing this "infinite sum" of the areas of infinitesimally small rectangles. Mathematically, the definite integral is defined as the limit of a Riemann Sum.

<BlockMath math="\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x" />

This notation has a specific meaning:
- **<InlineMath math="\int_{a}^{b}" />**: This is the integral symbol with a **lower** limit <InlineMath math="a" /> and an **upper** limit <InlineMath math="b" />. These numbers define the interval over which we are calculating the area.
- **<InlineMath math="f(x)" />**: This is the **integrand**, which is the function whose curve we are finding the area under.
- **<InlineMath math="dx" />**: Just as with indefinite integrals, this indicates that we are integrating with respect to the variable <InlineMath math="x" />.

Unlike an indefinite integral, which results in a function (<InlineMath math="F(x) + C" />), the result of a definite integral is a **single number** that represents the net area under the curve from <InlineMath math="x=a" /> to <InlineMath math="x=b" />.

## Calculating the Definite Integral with Limits

Let's try to calculate the exact value of a definite integral using its limit definition, as in the reference image example.

**Problem:** Determine the value of <InlineMath math="\int_{0}^{7} x \, dx" />.

**Solution:**

To solve this, we will convert it back to the limit form of a Riemann Sum.

**Step 1: Determine the Riemann Sum components**

From the problem <InlineMath math="\int_{0}^{7} x \, dx" />, we know:
- Function: <InlineMath math="f(x) = x" />
- Interval: <InlineMath math="[a, b] = [0, 7]" />

Thus, the width of each subinterval is:

<BlockMath math="\Delta x = \frac{b-a}{n} = \frac{7-0}{n} = \frac{7}{n}" />

We will use the right endpoint (<InlineMath math="x_i" />) as the sample point for each partition:

<BlockMath math="x_i = a + i \Delta x = 0 + i \left(\frac{7}{n}\right) = \frac{7i}{n}" />

**Step 2: Set up the Riemann Sum**

The height of each rectangle is <InlineMath math="f(x_i)" />, so:

<BlockMath math="f(x_i) = x_i = \frac{7i}{n}" />

Now, we plug this into the Riemann Sum formula:

<BlockMath math="\sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left(\frac{7i}{n}\right) \left(\frac{7}{n}\right) = \sum_{i=1}^{n} \frac{49i}{n^2}" />

**Step 3: Simplify and use summation properties**

We can factor constants out of the sigma notation, as <InlineMath math="n" /> is treated as a constant in the sum from <InlineMath math="i=1" /> to <InlineMath math="n" />.

<BlockMath math="\frac{49}{n^2} \sum_{i=1}^{n} i" />

Next, we replace <InlineMath math="\sum_{i=1}^{n} i" /> with its series summation formula, <InlineMath math="\frac{n(n+1)}{2}" />, and then simplify the expression to make it easier to evaluate the limit.

<div className="flex flex-col gap-4">
<BlockMath math="\frac{49}{n^2} \cdot \frac{n(n+1)}{2}" />
<BlockMath math="= \frac{49n(n+1)}{2n^2}" />
<BlockMath math="= \frac{49(n+1)}{2n}" />
<BlockMath math="= \frac{49}{2} \left( \frac{n+1}{n} \right) = \frac{49}{2} \left( 1 + \frac{1}{n} \right)" />
</div>

The simplification steps above ensure we get the easiest form to evaluate the limit. First, we cancel <InlineMath math="n" /> from the numerator and denominator, then we split the fraction.

> To solve the limit of a Riemann Sum, we often need some common series summation formulas:
> - <InlineMath math="\sum_{i=1}^{n} i = \frac{n(n+1)}{2}" />
> - <InlineMath math="\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}" />
> - <InlineMath math="\sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2" />

**Step 4: Take the Limit**

Finally, we take the limit as <InlineMath math="n \to \infty" /> to find the exact area.

<div className="flex flex-col gap-4">
<BlockMath math="\int_{0}^{7} x \, dx = \lim_{n \to \infty} \frac{49}{2} \left( 1 + \frac{1}{n} \right)" />
<BlockMath math="= \frac{49}{2} \left( 1 + 0 \right)" />
<BlockMath math="= \frac{49}{2}" />
</div>

So, the exact area of the region under the curve <InlineMath math="f(x) = x" /> from <InlineMath math="x=0" /> to <InlineMath math="x=7" /> is **49/2** or **24.5**. This is an example of how the definite integral provides an exact answer, no longer an approximation.