Ximpledu

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
To find the area under a curve y = f(x):
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 = x2 - 2x, x = 3, and the x-axis.
Solution

Example

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

Area between Two Curves

Formula


A = ab[f(x) - g(x)] dx
Top function: y = f(x)
Bottom function: y = g(x)

General formula: ∫ |f(x) - g(x)| dx

Definite Integral

Example

Find the area bounded by
y = x3, y = x2 - x, and x = 2.
Solution

Example

Find the area bounded by
y = x2 - 3x and y = x.
Solution