Evaluating Definite Integrals?

📌 CONCEPT: The definite integral ∫[−1, 1] √(x^2 + 1) dx represents the area of the region bounded by the curve y = √(x^2 + 1), the x-axis, and the lines x = 1 and x = -1. It evaluates the area under the curve in the specified interval. This integral is a classic example of evaluating the area between a curve and the x-axis within a given interval.

📐 RULE / FORMULA: To evaluate this integral, we use the antiderivative of √(x^2 + 1) which is ln(x + √(x^2 + 1)) + C, where C is the constant of integration. We then apply the Fundamental Theorem of Calculus (FTC) to find the definite integral by evaluating the antiderivative at the limits of integration.

💡 WORKED EXAMPLE: Let's find the area under the curve y = √(x^2 + 1) between x = -1 and x = 1. We evaluate the antiderivative ln(x + √(x^2 + 1)) at the limits of integration: ln(1 + √(1^2 + 1)) - ln(-1 + √((-1)^2 + 1)) = ln(2) - ln(i√2) = ln(2) - ln(√2) = ln(2) - ln(2)^(1/2) = ln(2) - ln(2)^(1/2).

⚠️ COMMON MISTAKE: Students often forget to apply the FTC correctly, neglecting to evaluate the antiderivative at the specified limits of integration, or incorrectly computing the antiderivative itself. They might also mistakenly calculate the area under the curve without considering the absolute value of the function when dealing with negative values of x.