Riemann Sums Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Riemann Sums.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Imagine filling the area under a curve with thin rectangles. The more rectangles you use, the better the approximation. In the limit of infinitely many infinitely thin rectangles, you get the exact area—which is the definite integral.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A Riemann sum estimates a definite integral by slicing $[a,b]$ into strips and adding the areas of rectangles whose heights come from the function.

Common stuck point: The procedure for riemann sums is the easy part; the trap is forgetting $\Delta x=\frac{b-a}{n}$ . Asking "Am I summing rectangle areas $f(x_i^*)\Delta x$ to estimate area under a curve, rather than evaluating exactly?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I summing rectangle areas $f(x_i^*)\Delta x$ to estimate area under a curve, rather than evaluating exactly?

Worked Examples

Example 1

easy

Approximate

\int_0^2 x^2\,dx

using a left Riemann sum with

n = 4

equal subintervals.

Answer

L_4 = 1.75

(underestimate; exact

= \frac{8}{3}

)

First step

\Delta x = 0.5

. Left endpoints:

x = 0, 0.5, 1, 1.5

Full solution

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.

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

Example 2

medium

Approximate

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

using a right Riemann sum with

n = 4

subintervals and classify the estimate.

Example 3

medium

Approximate

\int_0^4 x^2\,dx

using a midpoint sum with

n=4

Example 4

medium

Use a right Riemann sum with

n=4

to approximate

\int_0^2 e^x\,dx

. Round to 3 decimals.

Example 5

hard

Express

\int_0^2 x^3\,dx

as the limit of a right Riemann sum, then evaluate.

Example 6

hard

Use the limit of a right Riemann sum to show

\int_0^1 (3x+1)\,dx = 5/2

Example 7

medium

Approximate

\int_0^{\pi} \sin x\,dx

using a midpoint sum with

n=2

Example 8

medium

A car's velocity (m/s) over

[0,8]

s is sampled every 2 s:

v(0)=0

v(2)=10

v(4)=18

v(6)=24

v(8)=28

. Estimate distance traveled with a left Riemann sum.

Example 9

challenge

Recognize

\displaystyle\lim_{n\to\infty} \sum_{i=1}^{n} \frac{1}{n}\sqrt{1 + \frac{i}{n}}

as a definite integral and evaluate.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Use a midpoint sum with

n=2

to approximate

\int_0^4 (x+1)\,dx

Example 2

hard

Write the right Riemann sum for

\int_0^1 x^3\,dx

with

n

subintervals as a sigma expression and evaluate the limit.

Example 3

easy

For

\int_0^4 f(x)\,dx

with

n=4

subintervals, what is

\Delta x

Example 4

easy

For

\int_1^7 f(x)\,dx

with

n=3

, find

\Delta x

Example 5

easy

Estimate

\int_0^2 x\,dx

using a right Riemann sum with

n=2

Example 6

easy

Estimate

\int_0^2 x\,dx

using a left Riemann sum with

n=2

Example 7

easy

What does a Riemann sum approximate as

n\to\infty

Example 8

easy

For

\int_0^6 f(x)\,dx

with

n=6

, list the right endpoints.

Example 9

easy

Estimate

\int_0^4 (x+1)\,dx

with a left sum,

n=2

Example 10

easy

Which generally gives a better estimate for a fixed

n

: a left/right sum or the midpoint sum?

Example 11

medium

Estimate

\int_0^4 x^2\,dx

using a right Riemann sum with

n=4

Example 12

medium

Estimate

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

using a midpoint sum with

n=2

Example 13

medium

Estimate

\int_0^4 x^2\,dx

using the trapezoidal rule with

n=4

Example 14

medium

A car's speed (m/s) is recorded every 2 s:

t=0{:}0,\ 2{:}10,\ 4{:}20,\ 6{:}30

. Estimate distance with a right sum.

Example 15

medium

For

\int_0^1 f(x)\,dx

, write the right Riemann sum with general

n

as a sigma expression for

f(x)=x

Example 16

medium

Estimate

\int_0^3 (2x)\,dx

with a left sum,

n=3

, and compare to the exact value.

Example 17

challenge

Evaluate

\int_0^1 x^2\,dx

from the definition using the limit of right Riemann sums.

Example 18

challenge

Estimate

\int_0^2 e^x\,dx

using the trapezoidal rule with

n=2

and compare to the exact value

e^2-1

Example 19

challenge

For

\int_0^1 (3x^2)\,dx

, show the limit of right Riemann sums equals 1.

Example 20

medium

Estimate

\int_0^3 x^2\,dx

using a left Riemann sum with

n=3

Example 21

medium

Estimate

\int_1^5 \frac{1}{x}\,dx

using a right sum with

n=4

Example 22

medium

Estimate

\int_0^2 (x^2+1)\,dx

using a midpoint sum with

n=2

Example 23

easy

Compute

\Delta x

for

\int_2^{10} f(x)\,dx

with

n=4

subintervals.

Example 24

easy

List the left endpoints used for a left Riemann sum on

[0,6]

with

n=3

Example 25

easy

List the midpoints used for a midpoint Riemann sum on

[0,6]

with

n=3

Example 26

medium

For

\int_0^3 (4-x^2)\,dx

with

n=3

, find the left Riemann sum.

Example 27

medium

For

\int_0^4 \sqrt{x}\,dx

n=4

, find the midpoint sum (3 decimal places).

Example 28

medium

Approximate

\int_1^2 \frac{1}{x}\,dx

with the midpoint rule,

n=2

Example 29

medium

For an increasing function on

[a,b]

, which Riemann sum is an upper bound: left, right, or midpoint?

Example 30

easy

Write the right Riemann sum for

\int_a^b f(x)\,dx

with

n

subintervals in sigma form.

Example 31

medium

A table gives

f(0)=2

f(1)=3

f(2)=5

f(3)=4

f(4)=1

. Use a left Riemann sum,

n=4

, to estimate

\int_0^4 f(x)\,dx

Example 32

medium

Same table as the previous question. Find the right Riemann sum for

\int_0^4 f(x)\,dx

with

n=4

Example 33

medium

L_n

and

R_n

are left/right Riemann sums and

f

is decreasing on

[a,b]

, which is the upper bound?

Example 34

hard

For

\int_0^1 x^2\,dx

, evaluate the right Riemann sum as a function of

n

and take the limit.

Example 35

easy

Compute

\int_{-2}^{2} (2x+1)\,dx

approximately with a midpoint sum,

n=2

Example 36

medium

Same velocity data. Estimate distance with a right Riemann sum.

Example 37

hard

For

f(x) = x^2

[0,2]

with

n

subintervals, write

R_n

in closed form.

Example 38

medium

Approximate

\int_0^4 (5-x)\,dx

using a left Riemann sum with

n=4

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Related Concepts

Integral Definite Integral

Background Knowledge

These ideas may be useful before you work through the harder examples.

integraldefinite integral