Definite integrals (analytical, area under/between curves): notes and practice questions

Find $\int x^2 \sin(2x) dx$.

Hence, find the exact value of $\int_0^{\frac{\pi}{4}} x^2 \sin(2x) dx$.

The function $f$ is defined by $f(x) = e^x \sinh x$, where $x \in \mathbb{R}$.

Find the Maclaurin series for $f(x)$ up to and including the $x^3$ term.

Hence, find an approximate value for $\int_0^1 e^{x^2} \sinh(x^2)dx$.

The function $g$ is defined by $g(x) = e^x \cosh x$, where $x \in \mathbb{R}$.

Show that $g''(x) = 2g'(x)$.

Hence, find the values of $g'''(0)$ and $g^{(4)}(0)$.

Using the result from part (c), find the Maclaurin series for $g(x)$ up to and including the $x^4$ term.

Hence, or otherwise, determine the value of $\lim_{x \to 0} \frac{2e^x \cosh x - 2 - 2x - 2x^2}{x^3}$.

The acceleration, $a \text{ ms}^{-2}$, of a particle moving in a straight line at time $t$ seconds, $t \ge 0$, is given by $a = -2v - 8$, where $v \text{ ms}^{-1}$ is the particle's velocity. At $t=0$, the particle is at the origin O and has an initial velocity $v_0 \text{ ms}^{-1}$, where $v_0 > 0$.

By solving an appropriate differential equation, show that the particle's velocity at time $t$ is given by $v(t) = (v_0 + 4)e^{-2t} - 4$.

The particle moves in the positive direction until it reaches its maximum displacement from O at time $T$. Show that $e^{2T} = \frac{v_0+4}{4}$.

Find an expression for the maximum displacement, $s_{\text{max}}$, in terms of $v_0$.

Let $v(T-k)$ represent the particle's velocity $k$ seconds before it reaches $s_{\text{max}}$, where $0 < k < T$. By using the result from part (b)(i), show that $v(T-k) = 4(e^{2k} - 1)$.

Similarly, let $v(T+k)$ represent the particle's velocity $k$ seconds after it reaches $s_{\text{max}}$. Deduce a similar expression for $v(T+k)$ in terms of $k$.

Hence, show that the speed of the particle $k$ seconds before it reaches $s_{\text{max}}$ is greater than or equal to its speed $k$ seconds after it reaches $s_{\text{max}}$.

By using an appropriate substitution, show that $\int \sin(\sqrt{x}) \, dx = 2\sin(\sqrt{x}) - 2\sqrt{x} \cos(\sqrt{x}) + C$.

The following diagram shows part of the curve $y = \sin(\sqrt{x})$ for $x \ge 0$.

The curve intersects the x-axis at $x_1, x_2, x_3, \dots$.

The nth x-intercept of the curve, $x_n$, is given by $x_n = n^2 \pi^2$, where $n \in \mathbb{Z}^+$.

Write down an expression for $x_{n+1}$.

The regions bounded by the curve and the x-axis are denoted by $R_1, R_2, R_3, \dots$ as shown on the diagram.

Calculate the area of region $R_n$.

Give your answer in the form $(an+b)\pi$, where $a, b \in \mathbb{Z}^+$.

Hence, show that the areas of the regions $R_1, R_2, R_3, \dots$ form an arithmetic sequence.

Consider the function $f$ defined by $f(x) = x^3 - 6x^2 + 8x$.

Find the $x$-intercepts of the graph of $y=f(x)$.

The graph of $y=f(x)$ for $0 \le x \le 4$ is shown below. The graph encloses two regions with the $x$-axis, shaded in the diagram.

Find the total area of the shaded regions.

The total surface area of a closed right cylinder is 8, equal to the total shaded area found in part (b). The cylinder has a height of $\frac{4-\pi}{\pi}$.

Find the radius, $r$, of the cylinder.

Hence, find the volume of the cylinder.

Find $\int x^2 \cos(2x) dx$.

Hence, find the exact value of $\int_0^{\frac{\pi}{4}} x^2 \cos(2x) dx$.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

A continuous random variable $Y$ has probability density function $g$ defined by

$g(y)=\begin{cases} \frac{1}{4c}, & -c \le y \le 3c \\ 0, & \text{otherwise} \end{cases}$

where $c$ is a positive real number.

(a) State $E(Y)$ in terms of $c$.

(b) Use integration to find $Var(Y)$ in terms of $c$.

Consider the functions $f(x) = \sin x$ and $g(x) = \cos 2x$, where $0 \le x \le \pi$.

The graphs of $f$ and $g$ are shown in the following diagram.

The graphs intersect at points P and Q. The region enclosed by the two graphs is shaded and labelled R.

(a) Find the $x$-coordinates of P and Q.

(b) Find the area of R.

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

Hence, find the value of $\int_1^8 \frac{5x-2}{\sqrt[3]{x}} \, dx$.

Expand and simplify $(1+a)^3$ in ascending powers of $a$.

By using a suitable substitution for $a$, show that $1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) = 8\cos^6(x)$.

Consider the function $g(x) = 4\sin(x)(1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) )$.

Show that $\int_0^p g(x) \, dx = \frac{32}{7}(1-\cos^7 p)$, where $p$ is a positive real constant.

It is given that $\int_p^{\frac{\pi}{2}} g(x) \, dx = \frac{1}{28}$, where $0 \le p \le \frac{\pi}{2}$. Find the value of $p$.

A particle moves in a straight line. Its velocity, $v \,\text{ms}^{-1}$, at time $t$ seconds is given by $v(t) = -t^3 + 6t^2 - 9t$, for $0 \le t \le 5$. The particle is at the origin at $t=0$.

The graph of $v$ is shown in the following diagram.

(a) Find the displacement of the particle from the origin at $t=3$.

(b) Find an expression for the acceleration of the particle.

(c) The particle is momentarily at rest at $t=0$ and again at $t=k$. Find the greatest speed of the particle in the interval $0 \le t \le k$.

(d) Find the greatest speed of the particle for $0 \le t \le 5$.

(e) Write down an expression that represents the distance travelled by the particle while its speed is increasing. Do not evaluate the expression.

Consider the expression $f(x) = \frac{1}{1+4x}$, where $|x| < \frac{1}{4}$.

(a) (i) Show that the first four terms in the binomial expansion of $f(x)$ are $1 - 4x + 16x^2 - 64x^3$.

(ii) Hence, find an approximation for $\int \frac{1}{1+4x} dx$.

(b) (i) Use your result from part (a)(ii) to find an approximate value for $\int_0^{1/8} \frac{1}{1+4x} dx$.

(ii) Hence, find an approximation for $\ln(1.5)$ in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$.

Consider the family of integrals defined by $I_n = \int x^n e^{-x} \, dx$ for $n \in \mathbb{N}_0$, where $\mathbb{N}_0 = \{0, 1, 2, ...\}$.

(a) By using integration by parts, show that $I_n = -x^n e^{-x} + n I_{n-1}$ for $n \ge 1$.

(b) Hence, find an explicit expression for $\int x^3 e^{-x} \, dx$.

(c) The region $R$ is enclosed by the graph of $y = x^{3/2} e^{-x/2}$ and the $x$-axis for $x \ge 0$. The region $R$ is rotated by $2\pi$ radians about the $x$-axis. Find the volume of the solid generated.

(d) Show that $\lim_{x \to \infty} x^n e^{-x} = 0$ for any $n \in \mathbb{N}$.

Consider the function $h(x) = x e^{-x}$.

(e) (i) Find the Maclaurin series for $h(x)$ up to and including the term in $x^4$.

(ii) Hence, find the value of the fourth derivative of $h(x)$ at $x=0$, i.e. $h^{(4)}(0)$.

A small reconnaissance drone is performing a vertical ascent and descent. Its vertical velocity, $v \text{ ms}^{-1}$, at time $t$ seconds, for $0 \le t \le 10$, is modelled by the function $v(t) = t \sin t - 2.5$.

The following diagram shows the graph of $v$.

(a) Find the smallest value of $t$ for which the drone is momentarily stationary.

(b) Find the total vertical distance travelled by the drone during the first 10 seconds.

(c) Find the vertical acceleration of the drone when $t=8$ seconds.

Consider the function $f$ defined by $f(x)=\frac{6x^2}{(x^3+1)^2}$ for $x \ge 0$. The graph of $f$ is shown in the following diagram.

Show that $f'(x)=\frac{12x(1-2x^3)}{(x^3+1)^3}$.

Find $\int f(x)dx$.

Consider a function $g(x)$ defined for $x \ge 0$. The derivative of $g$ is such that $g'(x) = f'(x)$, for all $x \ge 0$.

Let $R$ be the region enclosed by the graph of $f$, the graph of $g$, the line $x = 0$ and the line $x = 2$. The area of $R$ is $\frac{16}{3}$.

Find the two possible expressions for $g(x)$.