Assignments

Integration can be used for the calculation of an area. This means that the corresponding function has to be integrated, taking into account the limits of integration: the so-called lower and upper boundaries.
When integration is used to calculate the content of solids of revolution there are two formulas of interest: one for the rotation of a graph around the $X$ -axis, the other around the $Y$ -axis. Sometimes we need to do some preparations before the integral can be calculated.

1. Calculate the area which is enclosed by the graph of $y=x^2$ , the $X$ -axis and the line $x=3$ .

Solution

2. Calculate the are which is enclosed by $y=\displaystyle\frac{1}{x}$ , the lines $x=1$ , $x=3$ and $y=2$ .

Solution

3. Calculate the area which is enclosed by the lines $Q=0$ and $P=40$ and the graph of the quadratic function $P=Q^2+6Q$ . This assignment is relevant for the calculation of the Producer Surplus (see Bradley T. en Patton P., Essential Mathematics for Economics and Business, 2e ed., John Wiley, p. 418).

Solution

4. Calculate the area which is enclosed by the lines $Q=0$ and $P=20$ and the graph of the function $P=\displaystyle\frac{100}{Q+2}$ .
This assignment is relevant for the calculation of the Producer Surplus (see Bradley T. en Patton P., Essential Mathematics for Economics and Business, 2e ed., John Wiley, p. 416).

Solution

5. Calculate the area which is enclosed by the circle $x^2+y^2=R^2$ , that is a circle with center $(0,0)$ and radius $R$ .

Solution

6. Calculate the volume of the cylinder which is the result of revolving the line $y=R$ , $x\in[0,h]$ around the $X$ -axis.

Solution

7. Calculate the content of the cone which is the result of revolving the line $y=\displaystyle\frac{1}{2}x$ , $x\in[0,2]$ around the $X$ -axis.

Solution

8. Calculate the volume of the sphere with center $(0,0,0)$ and radius $R$ .

Solution

9. Calculate the volume of the paraboloid which is the result of revolving the parabola $y=x^2+1$ , $y\in[1,3]$ around the $Y$ -axis.

Solution

10. Suppose we have a sphere with radius $R$ . We drill a cylinder in this sphere. The axis of the cylinder goes through the center of the sphere (compare a corer). After the drill it appears that the cylinder has a height of $2a$ with $2a\leq{R}$ . Calculate the volume of the spherical shell that remains after the cylinder has been drilled.

Solution

Area and volume

Assignments

Contact

Tutor Math, Statistics or GMAT?

Language preference