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