Apply the definite integral to compute the area between curves (including curves that change relative order), the average value of a function, and kinematics quantities (displacement, distance, position) from velocity and acceleration

What this dot point is asking

QCAA wants you to apply the definite integral to compute geometric area (between a curve and the $x$-axis, or between two curves), the average value of a function on an interval, and kinematics quantities (displacement, distance, position from velocity). The dot point feeds the IA3 PSMT (which often models a real-world rate function) and the EA Paper 2 extended response.

Area under a curve

The signed area between $y = f(x)$ and the $x$-axis on $[a, b]$ is:

For geometric area (always non-negative), split the interval at any zero of $f$ on $(a, b)$ and take absolute values:

Area between two curves

If $f(x) \geq g(x)$ on $[a, b]$, the area between them is:

"Top minus bottom".

Procedure for area enclosed by two curves:

1. Find intersection points. Solve $f(x) = g(x)$.
2. Identify top and bottom in each sub-interval between intersections. Test a value inside.
3. Integrate top minus bottom on each sub-interval.
4. Sum the (positive) sub-integrals.

If the two curves cross within the interval of interest, the top-and-bottom roles swap and you must split.

Average value of a function

The average value of $f$ on $[a, b]$ is:

Geometrically: the height of the rectangle on $[a, b]$ whose area equals the integral. In modelling: the typical value of a continuously varying quantity over an interval.

Example. Average velocity from $t_1$ to $t_2$ for a particle with velocity function $v(t)$:

(Note: this is average velocity, not average speed; speed requires the integral of $|v|$.)

Kinematics: position from velocity and acceleration

Displacement from velocity

For a particle with velocity $v(t)$ from $t = t_1$ to $t = t_2$:

Displacement is signed; positive if net motion is in the positive direction, negative otherwise.

Distance travelled from velocity

Total distance is the integral of speed $|v(t)|$:

For a velocity that changes sign on the interval (the particle changes direction), find the zeros of $v$, split the interval at each, and sum the absolute values of the sub-integrals.

Position function from velocity

If $v(t) = \frac{dx}{dt}$ and $x(t_0) = x_0$ is the initial position:

For most QCE Methods problems, antidifferentiate $v(t)$ to get $x(t) = \int v \, dt + C$, then use the initial condition to find $C$.

Velocity and position from acceleration

If $a(t) = \frac{dv}{dt}$:

Two integrations with two initial conditions ($v_0$ and $x_0$) determine the position function from acceleration.

Worked example. Constant-acceleration motion

A particle has constant acceleration $a = 3$ m/s$^2$, starting from rest at $x = 0$. Find $v(t)$ and $x(t)$.

$v(t) = \int 3 \, dt = 3t + C_1$. With $v(0) = 0$, $C_1 = 0$. So $v(t) = 3t$ m/s.

$x(t) = \int 3 t \, dt = \frac{3 t^2}{2} + C_2$. With $x(0) = 0$, $C_2 = 0$. So $x(t) = \frac{3 t^2}{2}$ m.

This recovers the standard $x = \frac{1}{2} a t^2$ formula from constant-acceleration kinematics.

Worked example. Variable acceleration

A particle has acceleration $a(t) = 6 - 2t$ m/s$^2$, with $v(0) = 1$ m/s and $x(0) = 0$. Find $v(t)$ and $x(t)$.

$v(t) = \int (6 - 2t) \, dt = 6t - t^2 + C_1$. $v(0) = 1 \implies C_1 = 1$. So $v(t) = 6t - t^2 + 1$.

$x(t) = \int (6t - t^2 + 1) \, dt = 3 t^2 - \frac{t^3}{3} + t + C_2$. $x(0) = 0 \implies C_2 = 0$. So $x(t) = 3 t^2 - \frac{t^3}{3} + t$.

PSMT applications

The IA3 PSMT often presents a real-world rate function: water flow into a reservoir, drug clearance from blood, energy consumption over a day, traffic flow. Integration of the rate function over an interval gives the total accumulated quantity; division gives the average rate.

Typical PSMT moves:

1. Identify the rate function and its domain.
2. Integrate to get the total change.
3. Apply initial / final conditions to determine constants.
4. Compute averages, peak values, or time-to-reach thresholds.
5. Discuss the model's limitations and refinements (boundary cases, real-world constraints not in the model).

Common errors

Mixing displacement and distance. Displacement is the signed integral; distance is the absolute-value integral or the split-and-sum.

Forgetting to split at zeros of $v$ for distance. A particle that changes direction has distance greater than displacement. The split-and-sum is mandatory.

Top-bottom backwards. Picking the wrong "top" gives a negative area. Test a value in each sub-interval before integrating.

Forgetting the constant of integration in position-from-velocity. $x(t)$ has $C$, which must be determined from $x(t_0) = x_0$.

Average velocity vs average speed. Average velocity = displacement / time. Average speed = distance / time. They differ when the particle reverses direction.

Forgetting to divide by interval length for average value. $\bar f = \frac{1}{b-a} \int_a^b f \, dx$; the $\frac{1}{b-a}$ is essential.

In one sentence

The definite integral applies to compute area between curves (top minus bottom on each sub-interval between intersections), the average value of a function (integral divided by interval length), and kinematics quantities (signed displacement from $\int v \, dt$, total distance from $\int |v| \, dt$ via split-and-sum, position function from $\int v \, dt$ plus an initial condition); the PSMT and EA examine these applications most heavily in real-world contexts involving rate functions and accumulated change.