5.5 Introduction to integration questions for IB Maths AA

Find the equation of an appropriate regression line.

Explain why the regression line of $y$ on $x$ is not appropriate for this purpose.

The resort's season lasts 120 days. The number of visitors on day $t$ of the season is modelled by

Find the number of visitors on day 30, according to this model.

Hence estimate the number of hot drinks the café sells on day 30.

Let $D(t)$ be the estimated number of hot drinks sold on day $t$, found by substituting $V(t)$ into the regression line from part (a).

Write down an expression for $D(t)$.

Find the rate at which the estimated number of hot drinks sold per day is changing on day 30.

By integrating $D(t)$, estimate the total number of hot drinks sold by the café during the 120-day season.

The graph of the quadratic function $y=a{x}^2+bx+c$ passes through the points $(-1,2)$, $(0,1)$ and $(1,4)$.

Find the value of $a$, the value of $b$ and the value of $c$.

Find ${\displaystyle {\int }_{-1}^1(a{x}^2+bx+c)\mathrm{d}x}$ for these values of $a$, $b$ and $c$.

Now consider the quadratic $y=a{x}^2+bx+c$ whose graph passes through the points $(-h,p)$, $(0,q)$ and $(h,r)$, where $h>0$ and $p,q,r\in \mathbb{R}$.

Show that, for all values of $p$, $q$ and $r$, this system has the unique solution

Hence show that ${\displaystyle {\int }_{-h}^h(a{x}^2+bx+c)\mathrm{d}x=\frac{h}{3}(p+4q+r)}$.

It can be shown that the result of part (d) holds for any three equally spaced points $(m-h,p)$, $(m,q)$ and $(m+h,r)$. The rule ${\displaystyle {\int }_{m-h}^{m+h}f(x)\mathrm{d}x\approx \frac{h}{3}(f(m-h)+4f(m)+f(m+h))}$ is used to approximate integrals.

Use this rule with $m=0.5$ and $h=0.5$ to find an approximate value for ${\displaystyle {\int }_0^1{\mathrm{e}}^{x^2}\mathrm{d}x}$, and find the percentage error of this approximation.

Show that the rule in part (d) gives the exact value of ${\displaystyle {\int }_{-h}^{h}f(x)\mathrm{d}x}$ for any cubic function $f(x)=k{x}^3+a{x}^2+bx+c$.

A rule of the form ${\displaystyle {\int }_{-1}^{1}f(x)\mathrm{d}x\approx \alpha f(-1)+\beta f(0)+\gamma f(1)}$ is required to give the exact value when $f(x)=1$, $f(x)=x$ and $f(x)=x^2$.

Write down a system of three equations in $\alpha$, $\beta$ and $\gamma$, and solve it. Show that your answer agrees with part (d).

A different rule uses unequally spaced points: ${\displaystyle {\int }_0^{3}f(x)\mathrm{d}x\approx Af(0)+Bf(1)+Cf(3)}$. It is required to give the exact value when $f(x)=1$, $f(x)=x$ and $f(x)=x^2$.

Find the value of $A$, the value of $B$ and the value of $C$.

Determine whether the rule in part (h) gives the exact value of ${\displaystyle {\int }_0^3{x}^3\mathrm{d}x}$.

Explain briefly the significance of the value of $A$ found in part (h).

Find the value of ${\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}$ when $t=10$, and interpret your answer in this context.

Find the values of $t$ for which the volume of water in the reservoir is increasing.

Find the volume of water in the reservoir when $t=20$.

Find the greatest volume of water in the reservoir for $0\le t\le 20$. Justify your answer.

Find the greatest rate at which the volume of water is decreasing for $0\le t\le 20$.

Express $f^{\mathrm{\prime }}(x)$ in the form $ax+b{x}^n$, where $a,b,n\in \mathbb{Z}$.

The graph of $f$ passes through the point $(3,20)$.

Find $f(x)$.