# Riemann Sum

See how to use the Riemann sum

to find the area under the function.

2 examples and their solutions.

## Riemann Sum

### Definition

S

_{n}= ∑k = 1nf(x

_{k})⋅△x

And draw a rectangle in each piece.

2. Find the area of the kth rectangle.

f(x

_{k})⋅△x.

3. Add the areas of the rectangles.

∑ f(x

_{k})⋅△x

Sigma (Math)

S

_{n}is the Riemann sum.

S = limn → ∞∑k = 1nf(x

_{k})⋅△x

### Example

Find the area of the colored region

by using the Riemann sum.

Solution by using the Riemann sum.

a

_{n}= (kn)

^{2}⋅1n - [3]

S = limn → ∞∑k = 1n(kn)

^{2}⋅1n

= limn → ∞∑k = 1nk

^{2}n

^{2}⋅1n

= limn → ∞∑k = 1nk

^{2}n

^{3}

= limn → ∞1n

^{3}∑k = 1nk

^{2}

= limn → ∞1n

^{3}⋅n(n + 1)(2n + 1)6 - [4]

= limn → ∞n(n + 1)(2n + 1)6n

^{3}

= 26 - [5]

= 13

[1]

Each rectangle's right side x value:

1/n, 2/n, 3/n, 4/n, ... , n/n

→ kth rectangle's right side x value: k/n

1/n, 2/n, 3/n, 4/n, ... , n/n

→ kth rectangle's right side x value: k/n

[2]

kth rectangle

Width: k/n

Height: f(k/n) = (k/n)

Width: k/n

Height: f(k/n) = (k/n)

^{2}[3]

a

_{n}: Area of the nth rectangle[4]

[5]

Close

### Types of Riemann Sums

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(x

_{k})⋅△x

= ∫abf(x) dx

→ 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 limn → ∞∑k = 1nf(kn + 1)1n

limn → ∞∑k = 1nf(kn + 1)1n

x = kn + 1 - [1]

dx = 1n - [2]

1) k = 1

a = limn → ∞(1n + 1) - [3]

= 0 + 1 - [4]

= 1

2) k = n

b = limn → ∞(nn + 1) - [5]

= 1 + 1

= 2

= ∫12f(x) dx

x = kn + 1 - [1]

dx = 1n - [2]

1) k = 1

a = limn → ∞(1n + 1) - [3]

= 0 + 1 - [4]

= 1

2) k = n

b = limn → ∞(nn + 1) - [5]

= 1 + 1

= 2

= ∫12f(x) dx

[1]

Set x = k/n.

[2]

Differentiate both sides.

[3]

1/n + 1: [x = k/n + 1] when k = 1

[4]

[5]

n/n + 1: [x = k/n + 1] when k = n

Close