definite integral (Riemann integral)

Definite Integral (Riemann Integral) in Math

Definition

The definite integral, also known as the Riemann integral, is a fundamental concept in calculus that allows us to calculate the area under a curve. It is used to find the total accumulation of a quantity over a given interval. The definite integral is denoted by the symbol ∫ and is defined as the limit of a sum.

Knowledge Points

The definite integral contains the following key points:

1. Partitioning: The interval over which the integral is calculated is divided into smaller subintervals.
2. Riemann Sum: The sum of the areas of rectangles formed by the function and the subintervals.
3. Limit: As the number of subintervals approaches infinity, the width of each subinterval approaches zero, resulting in a more accurate approximation of the area under the curve.
4. Antiderivative: The function being integrated must have an antiderivative, which allows us to find the area under the curve.

Formula

The formula for the definite integral (Riemann integral) is expressed as:

∫[a, b] f(x) dx = lim(n→∞) Σ[i=1 to n] f(xi) Δx

where:

• [a, b] represents the interval over which the integral is calculated.
• f(x) is the function being integrated.
• dx represents the differential element.
• xi is a point within each subinterval.
• Δx is the width of each subinterval.

Application

To apply the definite integral formula, follow these steps:

1. Determine the interval [a, b] over which the integral is to be calculated.
2. Choose a partition by dividing the interval into smaller subintervals.
3. Select a point within each subinterval.
4. Calculate the function value at each selected point.
5. Multiply each function value by the width of the corresponding subinterval.
6. Sum up all the products obtained in step 5.
7. Take the limit as the number of subintervals approaches infinity to obtain the exact value of the definite integral.

Symbol

The symbol for the definite integral (Riemann integral) is ∫.

Methods

There are several methods for evaluating definite integrals, including:

1. Riemann Sums: Using the limit of a sum to approximate the integral.
2. Geometric Interpretation: Interpreting the integral as the area under a curve.
3. Fundamental Theorem of Calculus: Utilizing the relationship between derivatives and integrals to evaluate the integral.
4. Integration by Substitution: Substituting variables to simplify the integral.
5. Integration by Parts: Applying the product rule of differentiation in reverse.

Solved Examples

1. Example 1: Calculate the definite integral of f(x) = 2x + 3 over the interval [1, 4]. Solution: ∫[1, 4] (2x + 3) dx = [x^2 + 3x] from 1 to 4 = (4^2 + 3(4)) - (1^2 + 3(1)) = 25.

2. Example 2: Find the area under the curve y = x^2 between x = 0 and x = 2. Solution: ∫[0, 2] x^2 dx = [x^3/3] from 0 to 2 = (2^3/3) - (0^3/3) = 8/3.

Practice Problems

1. Calculate the definite integral of f(x) = 3x^2 + 2x - 1 over the interval [-2, 2].
2. Find the area under the curve y = sin(x) between x = 0 and x = π.

FAQ

Q: What is the definite integral (Riemann integral)? The definite integral, also known as the Riemann integral, is a mathematical concept used to calculate the area under a curve or the total accumulation of a quantity over a given interval.

Q: How is the definite integral (Riemann integral) represented? The definite integral is represented by the symbol ∫ and is written as ∫[a, b] f(x) dx, where [a, b] is the interval and f(x) is the function being integrated.

Q: What are the methods for evaluating definite integrals? Some common methods for evaluating definite integrals include Riemann sums, geometric interpretation, the fundamental theorem of calculus, integration by substitution, and integration by parts. These methods help simplify the integral and find its exact value.

Q: Can the definite integral be used to find the area under any curve? Yes, the definite integral can be used to find the area under any curve as long as the function being integrated has an antiderivative. The integral calculates the sum of infinitely many rectangles, resulting in an accurate approximation of the area.