Evaluate the limit of a sum involving sine functions by recognizing it as a Riemann sum and computing the corresponding definite integral.

Vault Of Euclid

Evaluating a Limit of a Sum Using a Definite Integral

The definite integral core

Evaluate \(\lim_{n\to\infty} \frac{1}{n} \left( \sin \frac{\pi}{n} + \sin \frac{2\pi}{n} + \dots + \sin \frac{n\pi}{n} \right)\).

The solution and a full step-by-step explanation are here. Sign in to read them.

What you need first

Definition of the definite integral as a limit of Riemann sums
Antiderivatives of basic trigonometric functions
Evaluating definite integrals using the Fundamental Theorem of Calculus

What this problem tests

Tests the ability to recognize a Riemann sum within a limit expression, identify the corresponding function and interval, and evaluate the resulting definite integral.

This is a classic calculus problem that tests the understanding of the definition of the definite integral as a limit of Riemann sums.

Evaluating a Limit of a Sum Using a Definite Integral

What you need first

What this problem tests

More on The definite integral