# Areas Between Curves and Applications of Integration

> Area between two curves, total change from a rate, and motion problems solved by integrating.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/areas-between-curves/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The garden bed
Grace is building a curved garden bed. The front edge follows a straight path, and the back edge follows a curved fence. She needs to know the area between them to order the right amount of soil.

The area between two curves is the area under the top curve **minus** the area under the bottom one. The same idea finds how much water flows into a dam from a flow-rate graph, or how far a rocket travels from its velocity.

**In short:** The area between two curves is $\int_a^b (\text{top} - \text{bottom})\,dx$, where $a$ and $b$ are the points where they meet. Integration also turns a rate into a total change, such as velocity into displacement.

## 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.

### Worked example: A frame
A photo frame is 30 cm by 20 cm outside, with a 24 cm by 14 cm opening. Frame area $= 600 - 336 = 264$ cm².

### 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.**

### Key idea: Top minus bottom
Area between two graphs = area under the top one minus area under the bottom one, between the points where they meet.

### Watch out: Which one is on top?
Test a point between the intersections. Subtracting the wrong way gives a negative area.

**Mind map**

```text
Applications of integration
  Between curves
    Find where they meet
    Top minus bottom
    Integrate between meeting points
  Total change
    Area under a rate graph
    ∫ rate dt
  Motion
    v = ∫ a dt
    x = ∫ v dt
    Distance: split where v = 0
```

### Year 11: Counting change from a graph
A rate-of-flow graph made of straight segments can be split into rectangles and triangles. Add the areas above the axis (flow in) and subtract the areas below the axis (flow out) to get the **net change**. Year 12 replaces the shapes with integrals.

### Year 12: Area between curves
If $f(x) \geq g(x)$ for $a \leq x \leq b$:

$$\text{Area} = \int_a^b \big(f(x) - g(x)\big)\,dx$$

**Example.** Area between $y = x + 2$ and $y = x^2$:

$$\int_{-1}^{2}\left(x + 2 - x^2\right)dx = \left[\frac{x^2}{2} + 2x - \frac{x^3}{3}\right]_{-1}^{2}$$

$$= \left(2 + 4 - \frac{8}{3}\right) - \left(\frac{1}{2} - 2 + \frac{1}{3}\right) = \frac{10}{3} + \frac{7}{6} = \frac{9}{2}$$

So the area is 4.5 square units. If the curves swap over inside the interval, split it at each crossing.

**Unknown parameter.** The area between $y = x^2$ and $y = kx$ (with $k > 0$) runs from $x = 0$ to $x = k$: $\displaystyle\int_0^k (kx - x^2)\,dx = \frac{k^3}{2} - \frac{k^3}{3} = \frac{k^3}{6}$. If the area is $\frac{9}{2}$, then $k^3 = 27$ and $k = 3$.

**Integrating along the $y$-axis.** When curves are easier to write as $x$ in terms of $y$, use right minus left: $\displaystyle\int_c^d (x_{\text{right}} - x_{\text{left}})\,dy$. For $y^2 = x + 4$ and the line $x = 0$: the curve is $x = y^2 - 4$, which is left of the $y$-axis between $y = -2$ and $y = 2$. Area $= \displaystyle\int_{-2}^{2} \big(0 - (y^2 - 4)\big)\,dy = \frac{32}{3}$.

**Shapes from tangents.** To find the area between a curve and its tangents, find each tangent's equation first, then use top minus bottom on each side of where the tangents cross.

### Year 12: Motion and total change by integrating
Integration reverses differentiation, so:

$$v = \int a\,dt \qquad x = \int v\,dt$$

Use the starting conditions to find each constant.

**Example.** A particle starts at the origin from rest with $a = 6t$ m/s². Then $v = 3t^2 + C$ and $v(0) = 0$ gives $v = 3t^2$. Next $x = t^3 + C$ and $x(0) = 0$ gives $x = t^3$. At $t = 2$ it is 8 m from the origin.

**Distance versus displacement.** $\displaystyle\int_a^b v\,dt$ gives the change in position. If the particle turns around, split at the times when $v = 0$ and add the absolute values to get the total distance.

**Total change.** If water flows into a dam at $R(t) = 200 - 4t$ ML per day, the amount in the first 10 days is

$$\int_0^{10}(200 - 4t)\,dt = \big[200t - 2t^2\big]_0^{10} = 1800 \text{ ML}$$

**Greatest displacement.** The particle is furthest in one direction when it stops and turns, so solve $v = 0$, then find $x$ there. For $v = 8 - 2t$ starting at the origin: $v = 0$ at $t = 4$, and $x = 8t - t^2$ gives $x(4) = 16$ m.

## Quick summary

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

## Quiz

Try these questions from the chapter quiz.

1. A 10 m by 8 m lawn has a 4 m by 3 m pond. What area of lawn is left?
   - A. 92 m²
   - B. 68 m²
   - C. 80 m²
   - D. 12 m²
   - Answer: B. 68 m². $80 - 12 = 68$.

2. Where do $y = x^2$ and $y = 4$ meet?
   - A. $x = 4$
   - B. $x = 2$ and $x = -2$
   - C. $x = 16$
   - D. $x = 0$
   - Answer: B. $x = 2$ and $x = -2$. $x^2 = 4$.

3. Where do $y = x^2$ and $y = 3x$ meet?
   - A. $x = 3$ only
   - B. $x = 0$ and $x = 3$
   - C. $x = 0$ only
   - D. $x = 9$
   - Answer: B. $x = 0$ and $x = 3$. $x^2 - 3x = 0$, so $x(x - 3) = 0$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/areas-between-curves/#quiz