Area Between Curves Math Example 4

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Find the area between

y = x^2

and

y = 2x

1
Intersections: $x^2=2x \Rightarrow x=0,2$ .
2
On $(0,2)$ : $2x > x^2$ (test $x=1$ : $2>1$ ).
3
$A = \int_0^2(2x-x^2)\,dx = [x^2-\frac{x^3}{3}]_0^2 = 4-\frac{8}{3} = \frac{4}{3}$ .

Answer

\frac{4}{3}

Find intersections, determine which curve is on top, integrate the difference.

About Area Between Curves

The area of the region enclosed between two functions $f(x)$ and $g(x)$ from $x = a$ to $x = b$ , computed as $A = \int_a^b |f(x) - g(x)|\,dx$ .

Find the area between [formula] and [formula] from [formula] to [formula].

Find the total area enclosed between [formula] and the [formula]-axis.

Find the area between [formula] and the [formula]-axis where the parabola is above the axis.