Definite integrals: notes and practice questions

(a) An architect is designing the entrance to a new tunnel. The shape of the entrance is modelled by a quadratic curve, with the base of the tunnel on the x-axis. The curve has end points (0, 6) and (10, 6), and its vertex is (5, 9). Distances are measured in metres.

The quadratic curve can be expressed in the form $y = ax^2 + bx + c$ for $0 \leq x \leq 10$.

(a.i) Write down the value of $c$.

(a.ii) Hence, form two equations in terms of $a$ and $b$.

(a.iii) Hence, find the equation of the quadratic curve.

(b) Calculate the area of the tunnel entrance.

A landscape architect is designing a decorative water channel. The cross-section of the channel is modelled by the function $f(x) = x^3 - 4x$, for $-2 \leq x \leq 2$. The shaded region, $R$, represents the cross-sectional area of the channel, bounded by the graph of $y = f(x)$ and the $x$-axis.

and below for x in [0,2].]

Write down an integral that represents the area of $R$.

Find the area of $R$.

The architect considers a modified design, where the cross-section is given by $g(x) = 0.5f(x+1)$.

On the following set of axes, the graph of $y = f(x)$ has been drawn. On the same set of axes, sketch the graph of $y = g(x)$.

The region $R$ (the original cross-section) is rotated through $2\pi$ radians about the $x$-axis to form a three-dimensional decorative element. Find the volume of this element.

The cross-section of a proposed tunnel entrance is modelled by a curve. The heights of the tunnel are measured at horizontal intervals and are given in the table below. All measurements are in cm.

Horizontal distance, x (cm)

0

10

20

30

40

50

60

Vertical distance, y (cm)

0

1.875

6.0

10.125

12.0

9.375

0

(a) Use the trapezoidal rule with $h = 10$ to find an approximation for the cross-sectional area of the tunnel entrance.

(b)

It is given that the equation of the curve is $y = 0.0225x^2 - 0.000375x^3$, for $0 \le x \le 60$.

(i) Write down an integral to find the exact cross-sectional area.

(ii) Calculate the value of the cross-sectional area to two decimal places.

(c) Find the percentage error in the area found using the trapezoidal rule. Give your answer to two decimal places.

Alex is preparing for a calculus exam and encounters a series of integrals. For each integral, identify whether it can be evaluated analytically using standard techniques (and if so, state the appropriate analytical method) or if it requires numerical approximation using technology (e.g., a GDC).

(a) $\int \cos(3x + 2)\,dx$

(b) $\int 2x e^{x^2}\,dx$

(c) $\int x \sin(x)\,dx$

(d) $\int e^{2x^2}\,dx$

(e) $\int_{1}^{2} \frac{4x^2}{x^3 + 3}\,dx$

(f) $\int_{1}^{4} 4\sin(x^2)\,dx$

The internal shape of a specialized chemical reactor vessel is formed by rotating the curve $y = 8 \ln x$, for $0 \le y \le 12$, about the $y$-axis, where $x$ and $y$ are measured in centimetres. The vessel contains a liquid to a height of $h$ cm.

Show that the volume of the liquid, $V$, in terms of $h$ is $V = 4\pi(e^{\frac{h}{4}} - 1)$.

Hence find the maximum capacity of the reactor vessel in cm$^3$. Give your answer to three significant figures.

(a) A designer is creating a prototype for a decorative vase. The cross-section of the vase can be modelled by the function $f(x) = x\sqrt{4-x^2}$, for $-2 \le x \le 2$.

Sketch the graph of $y = f(x)$ on the following pair of axes.

(b) The region enclosed by the graph of $y = f(x)$ and the x-axis is rotated $360^\circ$ about the x-axis to form the body of the vase.

(i) Write down an integral that represents the volume of this vase.

(ii) Calculate the value of this integral.

(c) The designer decides to create a new, larger version of the vase, $y = g(x)$, by applying the following transformations to the original cross-section $y = f(x)$:

• A horizontal stretch by a scale factor of 3, parallel to the x-axis.

• A vertical stretch by a scale factor of 0.75, parallel to the y-axis.

Find the volume of this new vase.

A landscape architect is designing a section of a park. The boundary of a planned pathway can be modelled by the line $y = -x + 8$ and a decorative flower bed by the curve $y = 0.5x^2 - 4x + 10.5$. These two features intersect at points $(1, 7)$ and $(5, 3)$, as shown in the following diagrams.

In diagram 1, the region enclosed by the line $y = -x + 8$, $x = 1$, $x = 5$ and the $x$-axis has been shaded.

Calculate the area of the shaded region in diagram 1.

In diagram 2, the region enclosed by the curve $y = 0.5x^2 - 4x + 10.5$, and the lines $x = 1$, $x = 5$ and the $x$-axis has been shaded.

Write down an integral for the area of the shaded region in diagram 2.

Calculate the area of this region.

Hence, determine the area enclosed between the line $y = -x + 8$ and the curve $y = 0.5x^2 - 4x + 10.5$.

A population of microorganisms grows according to the function $P(t) = t^3 - 9t^2 + 23t - 15$, where $P(t)$ is the population in thousands and $t$ is the time in hours, for $0 \le t \le 5$ hours.

The population is zero when $t=1$, $t=5$ and $t=a$ hours.

Find the value of $a$.

Calculate the total population growth (area enclosed by the curve and the $t$-axis) from $t=1$ hour to $t=a$ hours.

Another species of microorganism has its population modelled by $Q(t) = -2t^2 + 10t - 8$ (in thousands). The total population growth of species $P$ from $t=1$ to $t=b$ hours is equal to the total population growth of species $Q$ from $t=1$ to $t=b$ hours.

Find the value of $b$, where $1 < b < 3$.

A civil engineer is surveying a river to estimate its cross-sectional area for flow rate calculations. The depth of the river is measured at regular horizontal intervals across its width, as shown in the table below.

Use the trapezoidal rule with $h = 10$ m to find an approximation for the cross-sectional area of the river.

It is given that the equation of the curve modelling the riverbed cross-section is $y = 0.0005x^2(60-x)$, for $0 \le x \le 60$.

Write down an integral to find the exact cross-sectional area.

Calculate the value of the cross-sectional area to two decimal places.

Find the percentage error in the area found using the trapezoidal rule compared to the exact area.

The cross-section of a decorative garden bed is modelled by the curve $y = 2(x-1)(x-3)^2$, where $x$ and $y$ are measured in metres. The garden bed is built on flat ground, represented by the $x$-axis.

(a) Write down the $x$-intercepts of this curve.

(b) Write down a definite integral that represents the area of the cross-section of the garden bed.

(c) Find the value of this area.

The rate of profit, $P(t)$, in thousands of dollars per month, for a new product is modelled by the piecewise function

$P(t) = \begin{cases} P_1(t), & 0 \leq t \leq T \\ P_2(t), & t \geq T \end{cases}$

where $P_1(t) = -t^2 + 6t$ and $P_2(t) = -2t + 16$. For a smooth transition in the profit rate, it is required that $P_1(T) = P_2(T)$.

Find the value of $T$.

Show that $P_1'(T) = P_2'(T)$.

The total profit from the product at time $t=0$ is zero.

Find the time when the total profit returns to its initial position.

(a) (i) Consider the function $f(x) = x^3$. Find $f'(x)$.

(ii) The first section of the stone wall's profile is given by $y = f(x)$ for $0 \le x \le 0.5$. A straight glass panel is to be installed tangent to this section of the wall at the point where $x=0.5$. Find the equation of this tangent line.

The full profile of the stone wall, $F(x)$, is defined by:

$F(x) = \begin{cases} x^3 & 0 \le x \le 0.5 \\ 0.75x - 0.25 & 0.5 < x \le 1.0 \end{cases}$

A smaller, decorative stone insert is designed using a transformation of $F(x)$. The graph of $G(x)$ is obtained from the graph of $F(x)$ by:

• a stretch scale factor of $\frac{1}{2}$ in the $x$ direction,

• followed by a stretch scale factor of $\frac{1}{2}$ in the $y$ direction,

• followed by a translation of $0.5$ units to the right.

Point P lies on the graph of $F(x)$ and has coordinates $(1.0, 0.5)$. Point Q is the image of P under the given transformations and has coordinates $(q_x, q_y)$.

Find the value of $q_x$ and the value of $q_y$.

The piecewise function $G(x)$ is given by

$G(x) = \begin{cases} k(x) & c \le x \le d \\ mx + n & d < x \le q_x \end{cases}$

(c) Find

(i) an expression for $k(x)$.

(ii) the value of $d$.

(iii) the value of $n$.

(d) (i) Calculate the total area of the profile of the stone wall, enclosed by $y = F(x)$, the $x$-axis, and the line $x = 1.0$.

The decorative insert $G(x)$ is placed within the main wall profile $F(x)$. The region of the main wall profile that is not covered by the insert is to be painted a contrasting colour. This region is bounded by $y=F(x)$, the $x$-axis, and the lines $x=0$ and $x=1$, excluding the area under $G(x)$ from $x=0.5$ to $x=1.0$. Find the area of this region.

(a) A company's quarterly profit, in millions of dollars, is modelled by the function $P(x) = 0.5x^3 - 2.5x^2 + x + 4$, where $x$ represents the number of production batches (in hundreds) and $x \ge 0$. The company breaks even when $P(x) = 0$.

Given that the company experiences an initial profitable period, then breaks even, and then becomes profitable again, find the largest value of $x$ for which the company breaks even after its initial profitable period, denoted as $a$.

(b) Use your graphic display calculator to find the total profit (in millions of dollars) made by the company when the production batches range from $x = 2$ to $x = a$.

A landscape architect is designing a new public park. The northern boundary of the park is modelled by the function $g(x)$, and other boundaries are straight lines. The plan view of the park is shown in the following diagram, where both axes represent distance and are measured in metres.

The function $g(x)$ models the northern boundary of the park between points B and C and is given by

$g(x) = -\frac{1}{10}x^2 + 4x + 15$, for $0 \le x \le 40$.

(i) Find $g'(x)$.

(ii) Hence find the coordinates of the point on the park's northern boundary that is furthest north.

Point A has coordinates $(0, 0)$, point B has coordinates $(0, 15)$, point C has coordinates $(40, 15)$ and point D has coordinates $(40, 0)$.

(i) Write down the integral which can be used to find the area of the shaded region representing the park.

(ii) Find the area of the park.

(i) A landscaper uses the trapezoidal rule with 4 intervals to estimate the area of the park. Calculate this estimate.

(ii) Calculate the percentage error in the landscaper's estimate.

(iii) Suggest how the landscaper might be able to reduce the error whilst still using the trapezoidal rule.

A square-shaped meditation garden, PQRS, is to be built within the park. The side [PS] lies on the southern boundary (x-axis), and the side [QR] lies on the eastern boundary of the park ($x=40$). Point Q lies on the northern boundary curve $g(x)$.

(i) Find the x-coordinate of point P for the largest area of the meditation garden.

(ii) Find the largest area of the meditation garden.

The rate of change of a certain quantity, $Q$, with respect to time, $t$ (in minutes), is modelled by the function $\frac{dQ}{dt} = \frac{10}{5t-2}$.

(a) Find an expression for $Q(t)$ in terms of $t$, assuming an arbitrary constant of integration.

(b) Given that the model is valid for $t > \frac{2}{5}$, find the exact total change in the quantity $Q$ from $t=1$ minute to $t=4$ minutes. Give your answer in the form $a \ln b$, where $a, b \in \mathbb{N}$.