Riemann Sums Math Example 2

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

Example 2

medium

Approximate

\int_1^3 \frac{1}{x}\,dx

using a right Riemann sum with

n = 4

subintervals and classify the estimate.

Solution

1
$\Delta x = 0.5$ . Right endpoints: $x = 1.5, 2, 2.5, 3$ .
2
$f(1.5) \approx 0.667,\; f(2)=0.5,\; f(2.5)=0.4,\; f(3) \approx 0.333$ .
3
$R_4 = 0.5(0.667+0.5+0.4+0.333) = 0.5 \times 1.9 = 0.95$ .
4
$f$ is decreasing, so right endpoints give minima: underestimate. Exact: $\ln 3 \approx 1.099$ .

Answer

R_4 = 0.95

(underestimate; exact

= \ln 3 \approx 1.099

)

For a decreasing function, right endpoints use the smallest value in each subinterval, giving an underestimate.

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 1 easy

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

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