Step-by-step: Calculate the area under the curve of a given function, such as f(x) = x^n, using definite integrals.

Calculating Area Under the Curve using Definite Integrals

Given a function f(x) = x^n, we want to find the area under its curve from a to b. This is done using definite integrals.

Step 1: Understand Definite Integrals A definite integral, denoted as ∫ from a to b f(x) dx, represents the signed area between the curve of f(x) and the x-axis from a to b.

Step 2: Find the Indefinite Integral (Antiderivative) First, find the antiderivative of f(x), denoted as F(x). For f(x) = x^n, the antiderivative is F(x) = (x^(n+1))/(n+1), where n ≠ -1.

Step 3: Apply the Fundamental Theorem of Calculus The Fundamental Theorem of Calculus states that the definite integral of a function can be evaluated using its antiderivative. So, we have:

∫ from a to b x^n dx = F(b) - F(a) = [(b^(n+1))/(n+1)] - [(a^(n+1))/(n+1)]

Step 4: Calculate the Area Now, plug in the values of a and b to find the area under the curve:

Area = [(b^(n+1))/(n+1)] - [(a^(n+1))/(n+1)]

For example, if f(x) = x^3 and we want to find the area under its curve from 0 to 2, we have:

Area = [(2^4)/(4)] - [(0^4)/(4)] = [16/4] - [0/4] = 4