If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Riemann sums review

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.A (LO)
,
LIM‑5.A.1 (EK)
,
LIM‑5.A.2 (EK)
,
LIM‑5.A.3 (EK)
,
LIM‑5.A.4 (EK)
Review how we use Riemann sums and the trapezoidal rule to approximate an area under a curve.

## What are Riemann sums?

A Riemann sum is an approximation of the area under a curve by dividing it into multiple simple shapes (like rectangles or trapezoids).
In a left Riemann sum, we approximate the area using rectangles (usually of equal width), where the height of each rectangle is equal to the value of the function at the left endpoint of its base.
In a right Riemann sum, the height of each rectangle is equal to the value of the function at the right endpoint of its base.
In a midpoint Riemann sum, the height of each rectangle is equal to the value of the function at the midpoint of its base.
We can also use trapezoids to approximate the area (this is called trapezoidal rule). In this case, each trapezoid touches the curve at both of its top vertices.
For each type of approximation, the more shapes we use, the closer the approximation would be to the actual area.
Resources differ on this point, but we call any approximation that uses rectangles a Riemann sum, and any approximation that uses trapezoids a trapezoidal sum.

## Practice set 1: Approximating area using Riemann sums

Problem 1.1
Approximate the area between the x-axis and f, left parenthesis, x, right parenthesis from x, equals, 0 to x, equals, 8 using a right Riemann sum with 3 unequal subdivisions.
x0348
f, left parenthesis, x, right parenthesis25711
The approximate area is
unitssquared.

Want to try more problems like this? Check out this exercise.

## Practice set 2: Approximating area using the trapezoidal rule

Problem 2.1
Approximate the area between the x-axis and h, left parenthesis, x, right parenthesis from x, equals, 3 to x, equals, 11 using a trapezoidal sum with 4 equal subdivisions.
x357911
h, left parenthesis, x, right parenthesis364812
The approximate area is
unitssquared.

Want to try more problems like this? Check out this exercise.