# Area under a Curve

See how to find the area under a curve y = f(x).

4 examples and their solutions.

## Area under a Curve

### Formula

A = ∫acf(x) dx + ∫cb[-f(x)] dx

= ∫ab|f(x)| dx

integral f(x)

where the curve is above the x-axis,

and integral -f(x)

where the curve is below the x-axis.

(to make the sign of the integral value (+).)

Riemann Sum

### Example

Find the area bounded by

y = x

Solution y = x

^{2}- 2x, x = 3, and the x-axis.A = ∫02[-(x

^{2}- 2x)] dx + ∫23(x

^{2}- 2x) dx

= ∫02[-x

^{2}+ 2x] dx + ∫23(x

^{2}- 2x) dx

= [-13x

^{3}+ 2⋅12x

^{2}]02 + [13x

^{3}- 2⋅12x

^{2}]23 - [1]

= [-13x

^{3}+ x

^{2}]02 + [13x

^{3}- x

^{2}]23

= -13⋅2

^{3}+ 2

^{2}- [-130

^{3}+ 0

^{2}] + 13⋅3

^{3}- 3

^{2}- [13⋅2

^{3}- 2

^{2}]

= -83 + 4 - [0 + 0] + 3

^{2}- 3

^{2}- [83 - 4]

= -83 + 4 - 83 + 4

= 8 - 163

= 243 - 163

= 83

Close

### Example

Find the area bounded by

y = sin x (0 ≤ x ≤ 2π) and the x-axis.

Solution y = sin x (0 ≤ x ≤ 2π) and the x-axis.

A = ∫0πsin x dx + ∫π2π[-sin x] dx

= [-cos x]0π + [-(-cos x)]π2π - [1]

= [-cos x]0π + [cos x]π2π

= -cos π - (-cos 0) + [cos 2π - cos π]

= -(-1) - (-1) + [1 - (-1)] - [2]

= 1 + 1 + (1 + 1)

= 4

[1]

∫ sin x dx = -cos x

Definite Integral

Definite Integral

Close

## Area between Two Curves

### Formula

A = ∫ab[f(x) - g(x)] dx

Bottom function: y = g(x)

General formula: ∫ |f(x) - g(x)| dx

Definite Integral

### Example

Find the area bounded by

y = x

Solution y = x

^{3}, y = x^{2}- x, and x = 2.A = ∫02[x

^{3}- (x

^{2}- x)] dx - [1]

= ∫02[x

^{3}- x

^{2}+ x] dx

= [14x

^{4}- 13x

^{3}+ 12x

^{2}]02

= 14⋅2

^{4}- 13⋅2

^{3}+ 12⋅2

^{2}- [14⋅0

^{4}- 13⋅0

^{3}+ 12⋅0

^{2}]

= 14⋅16 - 13⋅8 + 2 - [0 - 0 + 0]

= 4 - 83 + 2

= 6 - 83

= 183 - 83

= 103

[1]

Top function: y = x

Bottom function: y = x

^{3}Bottom function: y = x

^{2}- xClose

### Example

Find the area bounded by

y = x

Solution y = x

^{2}- 3x and y = x.x

^{2}- 3x = x - [1]

x

^{2}- 4x = 0

x(x - 4) = 0 - [2]

x = 0, 4 - [3]

→ b = 4

[1]

To find the upper limit, b,

find the x value of the right intersecting point.

find the x value of the right intersecting point.

↓

A = ∫04[x - (x

^{2}- 3x)] dx - [4]

= ∫04[x - x

^{2}+ 3x] dx

= ∫04[4x - x

^{2}] dx

= [4⋅12x

^{2}- 13x

^{3}]04

= [2x

^{2}- 13x

^{3}]04

= 2⋅4

^{2}- 13⋅4

^{3}- [2⋅0

^{2}- 13⋅0

^{3}]

= 2⋅16 - 13⋅64 - [0 - 0]

= 32 - 32⋅23

= 32(1 - 23)

= 32⋅13

= 323

[4]

Top function: y = x

Bottom function: y = x

Bottom function: y = x

^{2}- 3xClose