The Definite Integral

The Definite Integral

The definite integral is a fundamental concept in calculus that calculates the exact accumulation of a function over an interval. It is used to find areas, volumes, work, and many other quantities in physics, engineering, and economics.

1. Definition of the Definite Integral

The definite integral of a function $f(x)$ over the interval $[a, b]$ is written as:

It represents the signed area under the curve $y = f(x)$ from $x = a$ to $x = b$, where:

• $f(x)$ is the function being integrated.
• $a$ is the lower limit of integration.
• $b$ is the upper limit of integration.
• $dx$ indicates integration with respect to $x$.

2. The Definite Integral as a Limit of Riemann Sums

The definite integral is defined as the limit of a Riemann sum:

where:

• $\Delta x = \frac{b-a}{n}$ is the width of each subinterval.
• $x_i^*$ is a sample point in the $i$th subinterval.
• $f(x_i^*)$ is the function value at $x_i^*$.

This sum approximates the area under the curve, and as $n \to \infty$, it gives the exact accumulated value.

3. The Fundamental Theorem of Calculus

The definite integral is evaluated using the Fundamental Theorem of Calculus (FTC):

where $F(x)$ is the antiderivative of $f(x)$, meaning $F'(x) = f(x)$.

This theorem connects differentiation and integration, making definite integrals easy to compute.

4. Properties of Definite Integrals

1. Reversing Limits:

   $$\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$$

2. Integral of a Constant:

   $$\int_{a}^{b} c \, dx = c(b - a)$$

3. Linearity:

   $$\int_{a}^{b} [f(x) \pm g(x)] \, dx = \int_{a}^{b} f(x) \, dx \pm \int_{a}^{b} g(x) \, dx$$

4. Additivity:

   $$\int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx = \int_{a}^{b} f(x) \, dx$$

5. Integral of an Odd Function (if $f(-x) = -f(x)$):

   $$\int_{-a}^{a} f(x) \, dx = 0$$

6. Integral of an Even Function (if $f(-x) = f(x)$):

   $$\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$$

5. Examples of Evaluating Definite Integrals

Example 1: Compute $\int_{1}^{3} (2x + 1) \, dx$

Step 1: Find the Antiderivative

Step 2: Apply the Fundamental Theorem

Step 3: Evaluate at Limits

Example 2: Compute $\int_{0}^{\pi} \sin x \, dx$

Step 1: Find the Antiderivative

Step 2: Apply the Fundamental Theorem

Step 3: Evaluate at Limits

6. Applications of the Definite Integral

1. Finding Areas under curves.
2. Calculating Volumes using solid of revolution methods.
3. Physics Applications:
   • Work done by a force: $W = \int F(x) \, dx$.
   • Distance traveled from velocity: $s = \int v(t) \, dt$.
4. Economics:
   • Consumer and producer surplus.
5. Probability:
   • Computing continuous probability distributions.

7. Summary

• The definite integral calculates the accumulated value of a function over an interval.
• It is defined as a limit of Riemann sums.
• The Fundamental Theorem of Calculus allows easy evaluation using antiderivatives.
• It has important properties and real-world applications in physics, economics, and probability.

The definite integral is one of the most powerful tools in calculus, linking differentiation and integration in a profound way.