Area under/between a curve, volumes of revolution: 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 mathematical model for a physical phenomenon involves the expression $\left(\frac{1}{u} + 2\right)^2$.

(a) Expand $\left(\frac{1}{u} + 2\right)^2$.

The rate of change of a certain quantity is given by $f'(x) = \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4$.

(b) Find the indefinite integral $\int \left( \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4 \right) dx$.

A designer is creating a custom-shaped container. The cross-sectional profile of the container is defined by the curve $y = \frac{1}{x+1} + 2$ for $1 \le x \le 3$. The container is formed by rotating this region $2\pi$ about the x-axis.

(c) Calculate the volume of the solid formed. Give your answer in the form $\frac{\pi}{4}(a + b \ln(c) )$, where $a, b, c \in \mathbb{Z}$.

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 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.

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 $y = 0.5x^2 - 4x + 10.5$.

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

(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.

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 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 $1 < b < 3$.

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 $a, b \in \mathbb{N}$.

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.

A landscape architect is designing a large decorative fountain for a new public park. The outer structure of the fountain consists of a cylindrical base topped by a conical section. The inner part of the fountain, which holds the water, is a hollow space created by rotating a specific curve around the vertical axis.

The shape of the inner hollow is based on a transformation of the graph $y = -x^3$. The curve that defines the profile of the inner hollow is given by $y = 40 - 0.064x^3$. This transformation involves a vertical translation of $a$ units and a stretch parallel to the x-axis with a scale factor of $b$.

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

(a.ii) Find the value of $b$.

The cylindrical base of the fountain has a radius of $15$ m and a height of $5$ m. The conical section on top has the same base radius of $15$ m and a height of $8$ m. The inner hollow, described by the curve $y = 40 - 0.064x^3$, extends from the base of the fountain ($y=0$) up to the total height of the outer structure.

Find the volume of the solid material that makes up the fountain (i.e., the volume of the outer structure minus the volume of the inner hollow).

(a) A landscape architect is designing a curved retaining wall for a garden. The height of the wall, $h$ metres, is measured at $1$ m intervals along its horizontal length, $x$ metres, from one end. The measurements are given in the table below.

Use the trapezium rule to estimate the cross-sectional area of the wall, giving your answer correct to $2$ decimal places.

(b) It is later discovered that the equation connecting $x$ and $h$ is $h(x) = 0.5x(8-x)$.

Calculate the true value of the cross-sectional area of the wall.

(c) Find the percentage error in your estimation of the area you made in part (a).

(d) Explain why, in this case, the trapezium rule underestimates the true value.

(a) An architect is designing a decorative archway for a park entrance. The cross-section of one half of the archway is modeled. The archway is symmetrical about the y-axis.

The architect models the base section of the archway as a straight line passing through the points $(0, 2)$ and $(2, 4)$, where all units are in metres.

Find the equation of the line passing through these two points.

(b) The architect initially models the curved upper section of the archway using the following measured points:

$(2, 4)$, $(4, 5)$, $(5.5, 3)$, and $(7, 0)$.

(i) Find the equation of the least squares regression quadratic curve for these four points.

(ii) By considering the gradient of this curve when $x = 2$, explain why it may not be a good model for the archway.

(c) The architect decides that a better model for the curved section would be a quadratic curve with a maximum point at $(4.5, 5.5)$ and that passes through the endpoint $(7, 0)$.

Find the equation of this new quadratic model.

(d) Believing this to be a better model for the archway, the architect wants to estimate the volume of the solid generated by rotating this half-archway about the x-axis.

(i) Write down an expression for this estimate of the volume as a sum of two integrals.

(ii) Find the value of this estimate.

(a) A designer is creating a decorative glass orb. The profile of the orb's cross-section is modelled by the curve $f(x) = \sqrt{R^2 - x^2}$ for $x \in [-R, R]$, where $R$ is the radius of the orb.

The designer rotates the curve $f(x)$ around the $x$-axis by $2\pi$ radians. State the geometrical name of the solid generated.

(b) Write down a definite integral, including limits, that represents the volume of this solid.

(c) If the radius of the glass orb is $R = 6$ cm, calculate the exact volume of the orb.

A drone is launched vertically upwards from a platform. Its vertical velocity, $v \text{ ms}^{-1}$, at time $t$ seconds, is given by the function:

$v = -3t^2 + 18t - 15$, for $t \ge 0$.

Find the times when the drone is momentarily at rest.

Find the magnitude of the drone's vertical acceleration at $t = 5$ seconds.

Find the greatest speed of the drone in the interval $0 \le t \le 5$.

The drone starts from an initial height of $10$ metres above the ground. Find an expression for the height of the drone, $h$ metres, above the ground at time $t$ seconds.

Find the total distance travelled by the drone in the interval $0 \le t \le 4$.