Riemann Sums Math Example 1

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

Example 1

easy

Approximate

\int_0^2 x^2\,dx

using a left Riemann sum with

n = 4

equal subintervals.

Solution

1
$\Delta x = 0.5$ . Left endpoints: $x = 0, 0.5, 1, 1.5$ .
2
$f(0)=0,\; f(0.5)=0.25,\; f(1)=1,\; f(1.5)=2.25$ .
3
$L_4 = 0.5(0+0.25+1+2.25) = 0.5 \times 3.5 = 1.75$ .
4
Exact value: $\frac{8}{3} \approx 2.667$ . The left sum underestimates because $f$ is increasing.

Answer

L_4 = 1.75

(underestimate; exact

= \frac{8}{3}

)

For an increasing function, left endpoints give the minimum in each subinterval, so the left Riemann sum underestimates. More subintervals improve accuracy.

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.

Learn more about Riemann Sums →

More Riemann Sums Examples

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].

Example 4 hard

Write the right Riemann sum for [formula] with [formula] subintervals as a sigma expression and eval

← All Riemann Sums Examples Riemann Sums Concept

Related Concepts

Integral Definite Integral