Week 10 Exercises
10.3 Sheet 19 Centroids of area
Exercise 1
Determine the positionns of the centroids of the uniform laminae represented by the following curves and indicated boundaries.
- \(y=x^2\), the \(x\)-axis and \(x=0, x=2\);
- \(y=2x^3\), the \(x\)-axis and \(x=1, x=2\);
- The portion of \(y=2x-x^2\) above the \(x\)-axis.
Exercise 2
Determine the position of the centroid of the figure bounded by \(y = e^{2x}\), the \(x-\)axis, the \(y-\)axis and the ordinate at \(x = 2\).
Exercise 3
Determine the position of the centroid of the figure bounded by the curve \(y = 5\sin(2x)\), the \(x-\)axis and the ordinates at \(x = 0\) and \(x = \pi/6\).
Exercise 4
A curve is given parametrically as \(x = at^2, y = 2at\) where \(t\) is a parameter and \(a\) is a constant. Determine the position of the centroid about the for figure bounded by the values \(t = 1\) and \(t = 2\).
[Solutions: 1. (i) \(\overline{x}=1.5\), \(\overline{y}=1.2\), (ii) \(\overline{x}=1.65\), \(\overline{y}=4.8\), (iii) \(\overline{x}=1\), \(\overline{y}=0.4\); 2. \(\overline{x}=1.5\), \(\overline{y}=13.6\); 3. \(\overline{x}=0.3\), \(\overline{y}=1.5\); 4. \(\overline{x}=2.66\), \(\overline{y}=1.61\);]
10.4 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. \(40.2\);]