Area between two graphs

By the end you can find the area between two graphs: a curve and a line, or two curves.

The line $y = x + 1$ meets the curve $y = x^2 - 4x + 5$ at $A$ and $B$ . Find the area of the region enclosed between them.

Sometimes the meeting points are given; often finding them is part (a).

Find where they meet: set the two equations equal and solve. These $x$ -values are your limits.
Decide which graph is on top between the limits. A sketch, or one test value in between, settles it.
Subtract first: integrate (top − bottom) in one go. This works because the area between them is (area under the top graph) − (area under the bottom graph), and you can do both in one integral.
Top limit minus bottom limit, as usual.

Bonus: top − bottom is positive even if the region dips below the $x$ -axis, so no splitting is needed.

Find the area of the region between $y = x^2$ and the line $y = 4$ .

Find the area of the region between the curve $y = x^2$ and the line $y = 9$ .

The line $y = x$ and the curve $y = x^2$ meet at $x = 0$ and $x = 1$ . Which integral gives the area between them?

A student finds the area between $y = 2x$ and $y = x^2$ . Which line has the first error?

What they ask