L3 Integration

On a sunny spring day, the temperature inside a glasshouse is modelled by

$ T = 18 - 8\cos\left(\frac{\pi t}{12}\right) $

where $T

The mean value of a function $f(t)$ for $a \le t \le b$ is

$ \frac{1}{b - a}\int_a^b f(t)\,dt $

(a) Find the mean temperature in the glasshouse between 6 am and 6 pm. Give your answer in exact form.

(b) At what times of the day is the temperature in the glasshouse equal to this mean temperature? Give your answers to the nearest minute.

You must use calculus and show the results of any integration needed to solve the problem.

A rugby ball is designed to be 29 cm long and to hold 4.5 litres of air. On the axes shown, in centimetres, the $x$-axis runs along the length of the ball through its centre, O.

At a distance $x

$ y = a\cos\left(\frac{\pi x}{29}\right) $

for $-14.5 \le x \le 14.5$, and $a$ is a positive constant. The volume of the ball is

$ V = \pi\int_{-14.5}^{14.5} y^2\,dx $

Find the width of the ball at its widest point. Give your answer correct to 3 significant figures. (1 litre = 1000 cm³.)

You must use calculus and show the results of any integration needed to solve the problem.

A fibreglass flagpole is 6 m long, and it is thicker at the bottom than at the top. At a distance of $x$ metres from the bottom, its mass per metre is

$ \lambda = \frac{6}{x + 2}\ \text{kg per metre} $

for $0 \le x \le 6$.

The mass of the flagpole is $M = \displaystyle\int_0^6 \lambda\,dx$ kg. Its balance point is $\bar{x}$ metres from the bottom, where

$ \bar{x} = \frac{1}{M}\int_0^6 x\lambda\,dx $

(a) Find the mass of the flagpole.

(b) How far from the bottom of the flagpole is its balance point?

Give your answers correct to 3 significant figures.

You must use calculus and show the results of any integration needed to solve the problem.

A small rocky island rises 30 m above sea level. The area enclosed by its contour line at a height of $h$ metres above sea level is

$ A(h) = 12\,000e^{-h/10}\ \text{m}^2 $

for $0 \le h \le 30$. The volume of the island above sea level is

$ V = \int_0^{30} A(h)\,dh $

(a) Find the volume of the island above sea level.

(b) Half of this volume lies below a certain height. Find this height.

Give your answers correct to 3 significant figures.

You must use calculus and show the results of any integration needed to solve the problem.

A sailing club's sign is cut from a flat sheet of aluminium in the shape of a sail. On the axes shown, in metres, the sign is the region between the axes and the curve $y = 1.5 - 1.5x^2$, for $0 \le x \le 1$.

The sign's area is $A = \displaystyle\int_0^1 y\,dx$, and its balance point is $(\bar{x}, \bar{y})$, where

$ \bar{x} = \frac{1}{A}\int_0^1 xy\,dx $

$ \bar{y} = \frac{1}{2A}\int_0^1 y^2\,dx $

Find the coordinates of the sign's balance point.

You must use calculus and show the results of any integration needed to solve the problem.