Areas Between Curves and Applications of Integration

Starting point: Years 9 and 10

Composite areas

To find an area with a piece cut out, subtract: area of the whole minus area of the hole.

Where two graphs meet

To find where $y=x^2$ meets $y=x+2$, set them equal:

$$x^2=x+2\Rightarrow x^2-x-2=0\Rightarrow (x-2)(x+1)=0$$

They meet at $x=-1$ and $x=2$, at the points $(-1,1)$ and $(2,4)$. Between these, the line is above the parabola (try $x=0$: line gives 2, parabola gives 0).

Total from a rate

If water flows into a tank at 5 L/min for 8 minutes, the total is $5\times 8=40$ L, the area under the rate graph. The same is true for any rate: area under a rate graph = total change.

Quick summary

• For a region with a hole, subtract the areas.
• Find where two graphs meet by setting them equal.
• Area between curves: ${\int }_a^b(\text{top}-\text{bottom})dx$.
• Area under a rate graph is the total change.
• $v=\int adt$ and $x=\int vdt$. Split at $v=0$ for total distance.

Quiz