Which method is used to approximate the definite integral ∫ f(x) dx over an interval?

Which method is used to approximate the definite integral ∫ f(x) dx over an interval?

• A. Antiderivative
• B. Fundamental theorem of calculus
• C. Riemann sums
• D. Use exact area under curve

Answer: (C)

Riemann sums. To approximate the definite integral, you partition the interval [a, b] into n subintervals, choose a sample point in each subinterval, and sum the rectangle areas f(xi*)Δx with Δx = (b − a)/n. As you increase n (making Δx smaller), the sum gets closer to the true area under the curve, and in the limit it equals the definite integral when f is integrable. This is the standard numeric method used when an exact antiderivative is not available. The exact value can also come from an antiderivative via the fundamental theorem (if an antiderivative exists), but that yields an exact result rather than an approximation. And “exact area under the curve” isn’t a method of approximation.

Riemann sums. To approximate the definite integral, you partition the interval [a, b] into n subintervals, choose a sample point in each subinterval, and sum the rectangle areas f(xi*)Δx with Δx = (b − a)/n. As you increase n (making Δx smaller), the sum gets closer to the true area under the curve, and in the limit it equals the definite integral when f is integrable. This is the standard numeric method used when an exact antiderivative is not available. The exact value can also come from an antiderivative via the fundamental theorem (if an antiderivative exists), but that yields an exact result rather than an approximation. And “exact area under the curve” isn’t a method of approximation.