- A measure of the effective voltage, \(M\) volts, in an electrical circuit is given by
$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$
where \(V\) volts is the voltage at time \(t\) seconds. Pairs of values of \(V\) and \(t\) are given in the following table.
| \(t\) | 0 | 0.25 | 0.5 | 0.75 | 1 |
| \(V\) | - 48 | 207 | 37 | - 161 | - 29 |
| \(V ^ { 2 }\) | | | | | |
Use the trapezium rule with five values of \(V ^ { 2 }\) to estimate the value of \(M\).
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Figure 1 shows part of a curve \(C\) with equation \(y = x ^ { 2 } + 3\). The shaded region is bounded by \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 3\). The shaded region is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
Using calculus, calculate the volume of the solid generated. Give your answer as an exact multiple of \(\pi\).