Week 9 Exercises
9.3 Sheet 17 More problems on Numerical Integration
Exercise 1
Use the Simpson’s rule with six strips to find an approximate value for \[ \int_0^{\pi/4}x^2\sin(2x)\ dx, \] giving your answer correct to four decimal places.
9.4 Sheet 18 Mean and RMS Values
Exercise 1
Determine the mean and RMS values for the following functions
- \(x^2-x\) from \(x=1\) to \(x=3\);
- \(x(3-x)\) from \(x=0\) to \(x=3\);
- \(3\sin(2t)\) from \(t=0\) to \(t=\pi/2\);
- \(2+3\sin(\theta)\) from \(\theta=0\) to \(\theta=\pi\);
- \(\sin(x)\) from \(x=0\) to \(x=\pi\) and from \(x=0\) to \(x=2\pi\);
- \(\cos^2(x)\) from \(x=0\) to \(x=2\pi\);
- \(\sin^2(x)\) from \(x=0\) to \(x=\pi/6\).
Exercise 2
Determine the average value of \(y = \sin(3x) + \cos(x)\) for values of \(x\) ranging from \(0\) to \(\pi/6\).
Exercise 3
Evaluate the mean values for
- \(1/\sqrt{16-2x^2}\) from \(x=0\) to \(x=2\);
- \(1/(1+x^2)\) from \(x=0\) to \(x=1\).
Exercise 4
Determine the RMS values for the following functions
- \(x(2-x)\) from \(x=0\) to \(x=2\);
- \(1+\sin(x)\) from \(x=0\) to \(x=2\pi\);
- \(3+2\cos(x)\) from \(x=0\) to \(x=2\pi\);
- \(100\sin(pt)+60\sin(3pt)\) from \(t=0\) to \(t=2\pi/p\).
Exercise 5
The instantaneous voltage in an AC (alternating current) circuit is given by \(v = 6 + 8\cos(\omega t)\). Determine the RMS value for the time period \(t = 0\) to \(t = 2\pi/\omega\).
Exercise 6
An alternating current is given by \(i = a\sin(\theta)\). Determine the RMS value of the current over a half wave and show that the ratio of the RMS value to the mean value is approximately \(1.11\).
[Solutions: 1. (i) \(\overline{y}=2.33\), \(\overline{y}_{\text{RMS}}=2.92\), (ii) \(\overline{y}=1.5\), \(\overline{y}_{\text{RMS}}=1.64\), (iii) \(\overline{y}=1.9\), \(\overline{y}_{\text{RMS}}=2.12\), (iv) \(\overline{y}=3.9\), \(\overline{y}_{\text{RMS}}=4.02\), (v) \(\overline{y}=0.64\), \(\overline{y}_{\text{RMS}}=0.707\), (vi) \(\overline{y}=0.51\), \(\overline{y}_{\text{RMS}}=0.61\), (vii) \(\overline{y}=0.087\), \(\overline{y}_{\text{RMS}}=0.1\); 2. \(\overline{y}=1.59\); 3. (i) \(\overline{y}=0.277\), (ii) \(\overline{y}=0.78\); 4. (i) \(\overline{y}_{\text{RMS}}=0.73\); (ii) \(\overline{y}_{\text{RMS}}=1.22\); (iii) \(\overline{y}_{\text{RMS}}=\sqrt{11}\); (iv) \(\overline{y}_{\text{RMS}}=82.5\); 5. \(\overline{v}_{\text{RMS}}=8.2\); 6. \(\overline{i}=2a/\pi\), \(\overline{i}_{\text{RMS}}=a/\sqrt{2}\), ratio \(=1.11\);]
9.5 Sheet 20 Volumes of Revolution
Exercise 1
Determine the volume of the solid of revolution obtained by rotating the curve \(y = x^2\) about
- The \(x-\)axis between the limits \(x = 1\) and \(x = 3\);
- The \(y-\)axis between the same limits.
Exercise 2
A cone is generated by rotating the line \(y = x/2\) about the \(x-\)axis from \(x = 0\) to \(x = 10\). Determine the volume of the cone.
Exercise 3
Determine the volumes of revolution formed by rotating about the \(x-\)axis the areas between the curves of
- \(y = x^2 - 2x\) and the \(x-\)axis;
- \(y = 3x - x^2\) and the \(x-\) axis.
Exercise 4
Determine the volume generated when the plane figure bounded by \(y = 5\cos(2x)\), the \(x-\)axis and the ordinates at values \(x = 0\) and \(x = \pi/4\) rotates about the \(x-\)axis for a complete revolution.
Exercise 5
The parametric equations of a curve are given as \(x = 6t^2, y = 2t - t^2\).Determine the volume generated when the plane figure bounded by the curve, the \(x-\)axis and the ordinates corresponding to \(t = 0\) and \(t = 2\) rotates about the \(x-\)axis
[Solutions: 1. \(152\); 2. \(126\); 3. (i) \(3.35\), (ii) \(25.4\); 4. \(30.8\); 5. \(40.2\);]