Riemann Sums Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

Write the right Riemann sum for

\int_0^1 x^3\,dx

with

n

subintervals as a sigma expression and evaluate the limit.

Solution

1
$\Delta x = \frac{1}{n}$ , right endpoints $x_i = \frac{i}{n}$ .
2
$R_n = \sum_{i=1}^{n}\left(\frac{i}{n}\right)^3 \frac{1}{n} = \frac{1}{n^4}\sum_{i=1}^{n} i^3$ .
3
Using $\sum_{i=1}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2$ : $R_n = \frac{(n+1)^2}{4n^2}$ .
4
$\lim_{n\to\infty} R_n = \frac{1}{4}$ .

Answer

\int_0^1 x^3\,dx = \frac{1}{4}

This confirms the definition of the definite integral as a limit of Riemann sums, recovering the FTC result

\left[\frac{x^4}{4}\right]_0^1 = \frac{1}{4}

About Riemann Sums

A method of approximating the definite integral $\int_a^b f(x)\,dx$ by dividing the interval $[a, b]$ into subintervals and summing the areas of rectangles (or trapezoids) whose heights are determined by the function.

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More Riemann Sums Examples

Example 1 easy

Approximate [formula] using a left Riemann sum with [formula] equal subintervals.

Example 2 medium

Approximate [formula] using a right Riemann sum with [formula] subintervals and classify the estimat

Example 3 easy

Use a midpoint sum with [formula] to approximate [formula].

← All Riemann Sums Examples Riemann Sums Concept

Related Concepts

Integral Definite Integral