12th Mathematics (Arts & Science) Part 2 Chapter 4 Solution (Digest) Maharashtra state board

Chapter 4 Definite Integration

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Project on Definite Integration

Definite integration is a fundamental concept in calculus that deals with calculating the total accumulation of a quantity, which can be represented as the area under a curve. Here’s a detailed explanation:

Basic Definition

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

$\int_a^b f(x) \, dx$

This expression calculates the net area under the curve $y = f(x)$ from $x = a$ to $x = b$.

Geometric Interpretation

• Area Under the Curve: For non-negative functions (i.e., $f(x) \geq 0$), the definite integral represents the area under the curve from $x = a$ to $x = b$.
• Net Area: If the function takes on both positive and negative values, the definite integral represents the net area, where areas below the x-axis (where $f(x) < 0$) are subtracted from areas above the x-axis (where $f(x) > 0$).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, providing a practical way to evaluate definite integrals:

1. First Part: If $F$ is an antiderivative of $f$ on $[a, b]$, then:

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

2. Second Part: If $f$ is continuous on $[a, b]$, then the function $F$ defined by:

   $F(x) = \int_a^x f(t) \, dt$

   is continuous on $[a, b]$, differentiable on $(a, b)$, and $F'(x) = f(x)$.

Evaluation Techniques

• Antiderivatives: Find an antiderivative $F(x)$ of $f(x)$, then apply the Fundamental Theorem of Calculus.

  Example: To evaluate $\int_1^4 3x^2 \, dx$, find the antiderivative $F(x) = x^3$, then calculate $F(4) - F(1)$:

  $\int_1^4 3x^2 \, dx = [x^3]_1^4 = 4^3 - 1^3 = 64 - 1 = 63$

• Numerical Methods: When an antiderivative is difficult or impossible to find, numerical methods such as the Trapezoidal Rule or Simpson's Rule can approximate the value of a definite integral.

  Example: Using the Trapezoidal Rule to approximate $\int_1^2 e^x \, dx$ with $n = 4$ subintervals.

Properties of Definite Integrals

• Linearity: $\int_a^b [c f(x) + d g(x)] \, dx = c \int_a^b f(x) \, dx + d \int_a^b g(x) \, dx$, where $c$ and $d$ are constants.

• Additivity: If $c$ is between $a$ and $b$:

  $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$

• Reversal of Limits: $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$

Applications

Definite integration has a wide range of applications in various fields:

• Physics: Calculating displacement, work done by a force, center of mass, etc.
• Engineering: Determining stress and strain, fluid dynamics, electrical circuits, etc.
• Economics: Finding consumer and producer surplus, accumulated profits, etc.
• Biology: Modeling population growth, understanding rates of reaction in biochemistry, etc.

Example Problems

1. Area under a curve: Find the area under $y = \sin(x)$ from $x = 0$ to $x = \pi$:

   $\int_0^\pi \sin(x) \, dx = [-\cos(x)]_0^\pi = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2$

2. Net area between curves: Find the net area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$:

   $\int_0^1 (x - x^2) \, dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1 = \left( \frac{1}{2} - \frac{1}{3} \right) - (0 - 0) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

In summary, definite integration is a powerful mathematical tool for calculating areas, accumulations, and other quantities across a defined interval. It is essential in both theoretical and applied disciplines.