Definite Integration (simple, GDC): notes and practice questions

(a) A scientist is studying the rate at which a certain chemical compound dissolves in a solution. The rate of dissolution, $R$, in grams per minute, at time $t$ minutes, is modeled by the expression $R(t) = \frac{4\sqrt{t} - 7}{\sqrt{t}}$.

This expression can be written in the form $A - Bt^p$, where $A$, $B$, and $p$ are constants. Write down the value of $p$.

(b) Hence, calculate the total amount of compound dissolved between $t=4$ minutes and $t=9$ minutes.

A scientist is studying the oscillation of a pendulum, and models its angular displacement using the function $h(t) = \cos t - 0.5$ radians, where $t$ is time in seconds, for $0 \le t \le \pi$.

Find the time(s) when the angular displacement is zero.

Determine the range of angular displacement of the pendulum.

Find an expression for the angular velocity, $h'(t)$.

Calculate the net change in a quantity related to the displacement over the time interval $0 \le t \le \pi$, represented by the definite integral $\int_0^\pi h(t) dt$.

Explain why the value found in part (b) does not represent the total magnitude of angular displacement from the equilibrium position accumulated over the interval $0 \le t \le \pi$.

Find the total magnitude of angular displacement from the equilibrium position accumulated over the interval $0 \le t \le \pi$.

Show that $(\sqrt{x} + \frac{1}{\sqrt{x}})^2$ can be written as $x + 2 + \frac{1}{x}$.

Hence, find the exact value of $\int_{1}^{e^2} (\sqrt{x} + \frac{1}{\sqrt{x}})^2 dx$.

The expression $\frac{2\sqrt[3]{x}-5}{\sqrt[3]{x}}$ can be written as $2-5x^{p}$. Write down the value of $p$.

Hence, find the value of $\int_{1}^{8} \left(\frac{2\sqrt[3]{x}-5}{\sqrt[3]{x}}\right) dx$.

A scientist is studying the profile of a specialized lens. The cross-sectional shape of the lens can be modelled by a curve defined by the function $f(x) = \frac{1}{\sqrt{25 - 9x^2}}$ for $0 \le x \le \frac{5}{6}$.

Calculate the exact area under the curve $f(x)$ from $x=0$ to $x=\frac{5}{6}$ by using a suitable substitution.

Use technology (GDC) to determine the value of the area calculated in part (a), correct to 3 significant figures.

(a) The signal strength of a drone, $S(t)$, at a horizontal distance $t$ km from its launch point, is modelled by the function $S(t) = 5e^{-0.2t^2}$. Calculate the total signal exposure, given by $E = \int_0^2 S(t) dt$, correct to $5$ significant figures.

(b) (i) Due to the nature of the signal, the function $S(t)$ is symmetric about $t=0$. Explain how the total signal exposure from $t=-2$ to $t=2$ relates to the value found in part (a).

(b) (ii) The drone's launch point is moved, and the new signal strength function is given by $S(t-1)$. Calculate the total signal exposure from $t=1$ to $t=3$ for this new function, correct to $5$ significant figures.

The height of a section of a roller coaster track is modelled by the function $h(x) = x(x^2 - 4)$, where $h$ is the height in meters and $x$ is the horizontal distance in hundreds of meters, for $x \in \mathbb{R}$.

(a) Find the coordinates of the points where the track is at ground level ($h=0$).

(b) Determine an expression for the rate of change of height with respect to horizontal distance, $h'(x)$.

(c) Hence find the horizontal distances $x$ of the turning points on the track.

(d) Find the value of $\int_{-2}^2 h(x) dx$.

(e) Explain why the value of the integral found in part (d) does not represent the total vertical distance covered by the track relative to the ground between $x = -2$ and $x = 2$.

(f) Find the total vertical distance covered by the track relative to the ground between $x = -2$ and $x = 2$.

The lifespan of a certain electronic component, in years, is modelled by a continuous random variable $T$, with probability density function $f$ given by

$f(t) = \begin{cases} \frac{3}{2}ct^2, & 0 \le t \le c \\ \frac{3}{2}c^2(2c-t), & c < t \le 2c \\ 0, & \text{otherwise} \end{cases}$

where $c > 0$.

Show that $c$ satisfies the equation $5c^4 = 4$.

Find the median of $T$.

(a) A particle moves along a straight line. Its velocity, $v$, in metres per second, at time $t$ seconds, is given by $v(t) = \frac{6\sqrt{t} + 5}{\sqrt{t}}$, for $t > 0$.

This expression can be written in the form $A + Bt^p$. Write down the value of $p$.

(b) Hence, find the displacement of the particle between $t=1$ and $t=4$.

A biologist is studying the growth rate of a bacterial colony. The rate of growth, $G$, in thousands of cells per hour, at time $t$ hours, is modelled by the expression $G(t) = \frac{5\sqrt[3]{t^2} + 9}{\sqrt[3]{t^2}}$.

(a) This expression can be written in the form $A + Bt^p$, where $A$, $B$, and $p$ are constants. Write down the value of $p$.

(b) Hence, find the total increase in the number of bacteria, in thousands, from $t=1$ to $t=8$.