Areas Between Curves

Level 1 - Calculus topic page in Applications Of Integrals.

Principle

Area between curves is found by integrating vertical or horizontal slices. The integrand is always a non-negative distance between the bounding curves on the chosen interval.

Notation

\(A\)

area between curves

\(f(x),g(x)\)

upper and lower functions in x-slices

\(a,b\)

intersection or boundary x-values

The Core Method

For vertical slices, subtract lower from upper.

Area between curves

\[A=\int_a^b(\text{upper}-\text{lower})\,dx\]

Find intersections first if the interval is not given. If the upper curve changes, split the integral.

Worked Cases

Question

Find the area between \(y=x\) and \(y=x^2\) from \(x=0\) to \(x=1\).

Answer

On \([0,1]\), \(x\ge x^2\), so the area is \(\int_0^1(x-x^2)dx\). Integrate: \(x^2/2-x^3/3\). Evaluate from \(0\) to \(1\): \(1/2-1/3=1/6\).

Examples

Question

Find where \(y=2x\) and \(y=x^2\) intersect.

Answer

Set the functions equal: \(2x=x^2\). Rearranging gives \(x^2-2x=0\), so \(x(x-2)=0\). The intersections occur at \(x=0\) and \(x=2\).

Improper Integrals

Volumes