Area Between Two Curves (Ex 11I)
Year 12 Mathematical Methods Unit 3 · Cambridge §11I · Mr Wong
Learning intentions.
- Find the area enclosed between two curves using $\displaystyle\int_a^b [\text{top}-\text{bottom}]\,dx$
- Find limits of integration by solving $f(x)=g(x)$
- Split at every cross-over point when the upper curve swaps
Key result
If $f(x) \ge g(x)$ on $[a,b]$, then $\displaystyle A = \int_a^b \bigl[f(x)-g(x)\bigr]\,dx$.
4-step recipe: (1) sketch · (2) intersections $f=g$ for the limits · (3) test which is on top · (4) integrate $\int (\text{top}-\text{bot})\,dx$.
Warm-up — finding intersections
1. Find the $x$-coordinates of the intersection points of each pair.
a $y=x+1$ and $y=x^2-x-2$
b $y=x^2$ and $y=2x$
c $y=4-x^2$ and $y=x+2$
d $y=x^2$ and $y=8-x^2$
Part A — Straight area-between-curves calculations
2. Find the area enclosed by $y = x+1$ and $y = x^2 - x - 2$.
3. Find the area enclosed by $y = 4 - x^2$ and $y = x+2$.
4. Find the area enclosed by $y = x^2$ and $y = x + 2$.
Area Between Curves — continued
Part B (curves that swap) · Part C (VCAA-style)
Part B — Curves that cross inside the region
5. Find the total area enclosed between $y = x^3$ and $y = x$ on $-1 \le x \le 1$. (Hint: where do they cross? Top function swaps at $x = 0$.)
6. Find the area enclosed by $y = \sin x$ and $y = \cos x$ on $\tfrac{\pi}{4} \le x \le \tfrac{5\pi}{4}$. Give an exact answer.
Part C — Application & VCAA-style
7. Find the area of the region bounded by $y = e^x$, $y = x$, the $y$-axis and the line $x = 1$. Give an exact answer.
8. The two parabolas $y = x^2$ and $y = 8 - x^2$ enclose a region. Sketch and find its area.
9. Extension. The line $y = mx$ cuts a chord from the parabola $y = 4 - x^2$, enclosing a region (above the line, below the parabola) of area $\tfrac{125}{6}$. Find $m$, with $m > 0$. (Hint: intersections at $x = \tfrac{-m \pm \sqrt{m^2+16}}{2}$, then integrate top − bottom; you should reach $(m^2+16)^{3/2}/6 = \tfrac{125}{6}$.)