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.
The definite integral contains the following key points:
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:
To apply the definite integral formula, follow these steps:
The symbol for the definite integral (Riemann integral) is ∫.
There are several methods for evaluating definite integrals, including:
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.
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.
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.