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.

  1. \(y=x^2\), the \(x\)-axis and \(x=0, x=2\);
  2. \(y=2x^3\), the \(x\)-axis and \(x=1, x=2\);
  3. 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

  1. The \(x-\)axis between the limits \(x = 1\) and \(x = 3\);
  2. 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

  1. \(y = x^2 - 2x\) and the \(x-\)axis;
  2. \(y = 3x - x^2\) 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. \(40.2\);]