91Ó°ÊÓ

The variables \(x\) and \(y\) are believed to satisfy a relationship of the form \(y=a b^{x}\), where \(a\) and \(b\) are constants. Show graphically that the values obtained in an experiment and shown in the table below do verify the relationship. From your graph calculate approximate values of \(a\) and \(b\). \(\begin{array}{cccccc}x & 1 & 2 & 3 & 4 & 5 \\ y & 14.1 & 15.8 & 17.8 & 19.9 & 22.4\end{array}\)

Apply Simpson's rule using five ordinates to find an approximate value of \(\int_{0}^{\pi} \sin ^{\frac{3}{2}} x d x\).

\(R\) is the region in the first quadrant bounded by the \(y\)-axis, the \(x\)-axis from 0 to \(\frac{1}{2} \pi\), the line \(x=\frac{1}{2} \pi\) and part of the curve \(y=(1+\sin x)^{\frac{1}{2}}\). (a) Show that, when \(R\) is rotated about the \(x\)-axis through four right angles, the volume of the solid formed is \(\frac{1}{2} \pi(\pi+2)\). (b) Use the trapezium rule with three ordinates to show that the area of \(R\) is approximately \(0.63 \pi\).

A bowl is formed by rotating about the \(y\)-axis that part of the \(x\)-axis between \(\quad x=0\) and \(x=2\) and that part of the curve \(\quad y=(x-2)^{2}\) between \(x=2\) and \(x=4 .\) Calculate the volume inside the bowl. Water is poured into the bowl at a constant rate of 20 cubic units per second. Find the rate at which the depth of water is increasing when the water level is \(0.25\) units above the base of the bowl.