JEE Maths Notes: Chapter on Definite Integrals

1. Introduction to Definite Integrals

The concept of definite integrals is a fundamental topic in calculus, used to calculate the area under a curve between two points. This section introduces the concept of integration, which is the reverse process of differentiation. A definite integral is represented as the integral of a function over a specific interval [a, b], and its value corresponds to the signed area under the curve of the function from x = a to x = b. The fundamental theorem of calculus, which links differentiation and integration, is also introduced in this section.

2. Definite Integral Definition

This subtopic covers the formal definition of the definite integral. The definite integral of a function f(x) over an interval [a, b] is denoted by:

∫ab f(x) dx

It is defined as the limit of a sum of areas of rectangles under the curve as the width of the rectangles approaches zero. The subtopic includes:

• Geometrical Interpretation: The integral represents the signed area between the curve y = f(x) and the x-axis, over the interval [a, b]. Positive areas above the x-axis and negative areas below the x-axis are considered.
• Fundamental Theorem of Calculus: The connection between differentiation and integration, where the definite integral of a function over an interval can be computed using its antiderivative.

3. Properties of Definite Integrals

This section explores the important properties of definite integrals, which help simplify the process of solving integral problems. Key properties include:

• Linearity: The definite integral of a sum of functions is the sum of the definite integrals of the individual functions.
• Reversal of Limits: Changing the limits of integration reverses the sign of the integral.
• Additivity: If the interval [a, b] is split into two subintervals [a, c] and [c, b], then the definite integral over [a, b] is the sum of the definite integrals over [a, c] and [c, b].
• Symmetry: For odd functions, the integral over a symmetric interval results in zero. Similarly, for even functions, the integral over a symmetric interval can be simplified by considering only half of the interval.

4. Techniques of Integration for Definite Integrals

In this subtopic, students learn various techniques to evaluate definite integrals. Some of the commonly used methods include:

• Substitution Method: When a function can be simplified by substituting a part of it with a new variable, the integral becomes easier to compute.
• Integration by Parts: A technique that uses the product rule of differentiation to split the integral of a product of two functions into simpler integrals.
• Trigonometric Substitution: A method for integrating certain types of functions involving square roots by using trigonometric identities.
• Partial Fraction Decomposition: Decomposing rational functions into simpler fractions to make the integration process easier.

5. Area Under the Curve

One of the primary applications of definite integrals is finding the area under a curve. This section explains how to compute the area between a curve y = f(x) and the x-axis over an interval [a, b]. The definite integral of f(x) over [a, b] gives the area under the curve, considering the sign of the function. The formula is:

A = ∫ab |f(x)| dx

For functions that dip below the x-axis, the absolute value is used to ensure that the area is always positive. This section also includes examples of areas under various types of curves.

6. Average Value of a Function

The average value of a function over an interval [a, b] is defined as the total area under the curve divided by the length of the interval. The formula for the average value of a function f(x) over [a, b] is:

Average Value = (1 / (b - a)) ∫ab f(x) dx

This concept is useful in physics and engineering for finding quantities like average speed, average temperature, and average concentration over a given period or range.

7. Applications of Definite Integrals

Definite integrals have numerous real-world applications in physics, engineering, economics, and biology. This section covers various practical uses, including:

• Finding Distances and Displacements: The definite integral is used to calculate the total distance traveled by an object moving along a path. If the velocity function is given, its integral gives the displacement over a specific time interval.
• Work Done by a Force: In physics, the work done by a variable force is calculated as the integral of the force function with respect to displacement.
• Volume of Solids: Definite integrals are used to calculate the volume of solids of revolution formed by rotating a curve about an axis. Methods such as the disk method and the shell method are explained in this section.
• Center of Mass: The center of mass of a system is determined using integrals to find the balance point of an object or distribution of mass.

8. Definite Integrals of Standard Functions

In this subtopic, students are introduced to the definite integrals of common functions that are frequently encountered in calculus problems. These include:

• Polynomials: The integral of polynomials over a given interval.
• Exponential Functions: How to integrate functions of the form e^x, a^x, etc.
• Trigonometric Functions: Definite integrals of sin(x), cos(x), sec(x), etc., over various intervals.
• Logarithmic Functions: The integration of logarithmic functions, particularly the natural logarithm.

This section includes worked-out examples for each type of function to help students understand the process and improve their problem-solving skills.

9. Integration of Piecewise Functions

Piecewise functions are those that are defined in different ways over different intervals. This section explains how to evaluate definite integrals of piecewise functions by splitting the integral into separate parts corresponding to the different intervals where the function is defined differently. Each piece is integrated individually, and the results are added together.

10. Numerical Integration

In some cases, the function may not have a simple antiderivative, and it may be difficult to evaluate the definite integral analytically. In such cases, numerical methods are used to approximate the value of the integral. This section introduces methods such as:

• Trapezoidal Rule: Approximating the area under the curve by dividing the interval into small trapezoids and summing their areas.
• Simpson’s Rule: A more accurate method for approximating integrals by using quadratic approximations to the function on each subinterval.