Chapter 26

Certain physical quantities are often represented as an area under a curve. By definition, power is the time rate of change of performing work. Thus, \(p=d w / d t\) or \(d w=p d t .\) If \(p=12 t-4 t^{2},\) find the work (in J) performed in 3 s by finding the area under the curve of \(p\) vs. \(t .\) See Fig. \(26.16 .\) Round the answer to three significant digits.

The total electric charge \(Q\) (in \(\mathrm{C}\) ) to pass a point in the circuit from time \(t_{1}\) to \(t_{2}\) is \(Q=\int_{t_{1}}^{t_{2}} i d t,\) where \(i\) is the current (in A). Find \(Q\) if \(t_{1}=1 \mathrm{s}, t_{2}=4 \mathrm{s},\) and \(i=0.0032 t \sqrt{t^{2}+1}\)

Because the displacement \(s,\) velocity \(v,\) and time \(t\) of a moving object are related by \(s=\int v d t\), it is possible to represent the change in displacement as an area. A rocket is launched such that its vertical velocity \(v\) (in \(\mathrm{km} / \mathrm{s}\) ) as a function of time \(t\) (in s) is \(v=1-0.01 \sqrt{2 t+1} .\) Find the change in vertical displacement from \(t=10 \mathrm{s}\) to \(t=100 \mathrm{s}\)

The total cost \(C\) (in dollars) of production can be interpreted as an area. If the cost per unit \(C^{\prime}\) (in dollars per unit) of producing \(x\) units is given by \(100 /(0.01 x+1)^{2}\), find the total cost of producing 100 units by finding the area under the curve of \(C^{\prime}\) vs \(x\).

A cam is designed such that one face of it is described as being the area between the curves \(y=x^{3}-2 x^{2}-x+2\) and \(y=x^{2}-1\) (units in \(\mathrm{cm}\) ). Show that this description does not uniquely describe the face of the cam. Find the area of the face of the cam, if a complete description requires that \(x \leq 1\)

A coffee-table top is designed to be the region between \(y=0.25 x^{4}\) and \(y=12-0.25 x^{4}\) (dimensions in \(\mathrm{dm}\) ). What is the area of the table top?