Definite Integrals

Definite integration is the process of calculating the exact area under a curve over a specific interval. Unlike indefinite integration, which results in a family of functions (antiderivatives) plus a constant of integration, definite integration yields a specific numerical value representing the net area under the curve.

1. The Definite Integral:

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The definite integral of a function $f(x)$ from a to $b$ is denoted by: $\int_a^b f(x) \, dx$ This represents the net area under the curve $y=f(x)$ from $x=a$ to $x=b$

2. Interpreting the Definite Integral:

Area Under the Curve: If $f(x)$ is positive over $\ [a, b]$ , the definite integral represents the area between the curve and the $x$ -axis from $x = a$ to $x = b.$
Net Area: If $f(x)$ takes on both positive and negative values over $\ [a, b],$ the definite integral gives the net area, with regions below the $x$ -axis subtracting from the total area.
Total Area: To find the total area regardless of whether the function is above or below the $x$ -axis, you would integrate the absolute value of the function.

3. Fundamental Theorem of Calculus:

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The Fundamental Theorem of Calculus connects differentiation and integration and provides a straightforward method to compute definite integrals: $\int_a^b f(x) \, dx = F(b) - F(a) \\$ where $F(x)$ is an antiderivative of $f(x)$ (i.e., $F'(x) = f(x)$ ).

4. Steps to Compute a Definite Integral:

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Find an Antiderivative $F(x)$ of $f(x$ ): Determine the indefinite integral $\int f(x) \, dx = F(x) + C.$
Evaluate the Antiderivative at the Limits: Compute $F(b) - F(a).$
Interpret the Result: The result $F(b)$ - $F(a)$ gives the net area under the curve from $x = a$ to $x = b$ .

5. Examples of Definite Integration:

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Example 1: Basic Polynomial Function

Compute $\int_1^3 (2x^2 + 3x) \, dx.$

Step 1: Find the antiderivative: $\int (2x^2 + 3x) \, dx = \frac{2x^3}{3} + \frac{3x^2}{2} + C \\$ So, $F(x) = \frac{2x^3}{3} + \frac{3x^2}{2}.$
Step 2: Evaluate at the limits: $F(3) - F(1) = \left(\frac{2(3)^3}{3} + \frac{3(3)^2}{2}\right) - \left(\frac{2(1)^3}{3} + \frac{3(1)^2}{2}\right) \\$ $F(3) = \frac{2(27)}{3} + \frac{27}{2} = 18 + 13.5 = :highlight[31.5]\\$ $F(1) = \frac{2(1)}{3} + \frac{3(1)}{2} = \frac{2}{3} + \frac{3}{2} = \frac{4 + 9}{6} = \frac{13}{6} \approx :highlight[2.167]\\$ $\text{Result: } :success[31.5 - 2.167 \approx 29.333]$

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Example 2: Trigonometric Function

Compute $\int_0^{\pi/2} \sin(x) \, dx.$

Step 1: Find the antiderivative: $\int \sin(x) \, dx = -\cos(x) + C \\$ So, $F(x) = -\cos(x).$
Step 2: Evaluate at the limits: $F\left(\frac{\pi}{2}\right) - F(0) = -\cos\left(\frac{\pi}{2}\right) - (-\cos(0))\\$ $F\left(\frac{\pi}{2}\right) = -\cos\left(\frac{\pi}{2}\right) = :highlight[0]\\$ $F(0) = -\cos(0) = :highlight[-1]\\$ $\text{Result: } :success[0 - (-1) = 1]$

6. Properties of Definite Integrals:

Definite integrals have several important properties that can simplify calculations:

Linearity: $\int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx$ $\int_a^b c \cdot f(x) \, dx = c \cdot \int_a^b f(x) \, dx$ where c is a constant.
Reversing Limits: $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$
Zero Integral: $\int_a^a f(x) \, dx = :highlight[0]$
Additivity: $\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx$

7. Applications of Definite Integrals:

Area under a Curve: The most common application, where the definite integral represents the area under the curve between two points.
Physics: Used to calculate quantities like displacement, work, and energy when integrating rates over time or other variables.
Economics: Used to find consumer and producer surplus, or to calculate total revenue or cost over a given interval.
Probability: Used in finding probabilities and expected values in continuous probability distributions.

Summary:

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Definite integration provides a way to calculate the exact area under a curve between two points, yielding a specific numerical result.
Definite integrals are widely used across mathematics, physics, economics, and many other fields to solve practical problems involving areas, accumulated quantities, and rates of change.