Understanding the Definite Integral: Calculating the Exact Area Under a Curve and its Applications

Definite Integral Definition

The definite integral is a fundamental concept in calculus that allows us to determine the exact value of the area under the curve of a function between two endpoints

The definite integral is a fundamental concept in calculus that allows us to determine the exact value of the area under the curve of a function between two endpoints. It can also be interpreted as finding the signed area.

The definite integral of a function f(x) over an interval [a, b] is denoted as ∫[a, b] f(x) dx. The integral sign (∫) represents the concept of integration, while the “a” and “b” represent the lower and upper limits of integration, respectively. The “dx” signifies that we are integrating with respect to the variable x.

Mathematically, the definite integral is defined as the limit of a sum. It can be calculated using the following formula:

∫[a, b] f(x) dx = lim(n→∞) Σ[f(xᵢ) Δxᵢ]

In this formula, n represents the number of subintervals (usually referred to as partitions) into which we divide the interval [a, b]. The values xᵢ represent the endpoints of each subinterval, and Δxᵢ represents the width of each subinterval (equal to (b-a)/n). The Σ symbol denotes the sum from i=1 to n.

To evaluate the definite integral, we need to find the antiderivative (also known as the indefinite integral) of the function f(x). The antiderivative is a function whose derivative recovers the original function. Once we find the antiderivative, we substitute the upper and lower limits of integration and subtract the values to obtain the final result.

In summary, the definite integral allows us to find the exact area under a curve by evaluating the antiderivative of the function between specified endpoints. It is a powerful tool in calculus for solving various mathematical and real-world problems.

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