Ximpledu

Riemann Sum

See how to use the Riemann sum
to find the area under the function.
2 examples and their solutions.

Riemann Sum

Definition

To find the area under y = f(x):

Sn = k = 1nf(xk)△x
1. Slice the area into n pieces.
And draw a rectangle in each piece.
2. Find the area of the kth rectangle.
f(xk)△x.
3. Add the areas of the rectangles.
f(xk)△x
Sigma (Math)
Sn is the Riemann sum.

S = limn → ∞k = 1nf(xk)△x
4. Set n → ∞.

Example

Find the area of the colored region
by using the Riemann sum.
Solution

Types of Riemann Sums

The Riemann sums can be different
by the heights of the rectangles.
Height: f(right x), f(middle x), f(left x)
Without the limit,
these Sums are not equal.
But, if n → ∞,
these Sums become equal.

Riemann Sum → Definite Integral

Formula


S = limn → ∞k = 1nf(xk)△x
= abf(x) dx
This is the way to change the limit of a Riemann sum to a definite integral.
→ Definite integral: The sum of the area of the thinly sliced rectangle.
(when y = f(x) is above the x-axis.)
a: x value when k = 1, n → ∞
b: x value when k = n, n → ∞

Example

Change the given expression to a definite integral.
limn → ∞k = 1nf(kn + 1)1n
Solution