Volume of Revolution

This chapter is about using integration to find the volume of a particular kind of solid, called a solid of revolution, When you have completed it, you should
be able to find a volume of revolution about either the x or y axis.

Examples of how the graphs look like :

When the region under the graph of y = f(x) between x = a and x = b ( where x = -1 and x = 1) is rotated about the x-axis, the volume of the solid of revolution formed is 

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Integration

Okay, i will skip the introducing part and go straight to the point. Basically you use integration to find the area and volume. Many of the formulae you have learnt, such as those for the volume of a sphere or a cone, can be proved by using integration. This chapter deals only with areas.

Ex 1 :
Find the area under y=1/x^2 from x = 2 and x = 5.



Some formulaes :







 The area between two graphs

You sometimes wan to find the area of a region bounded of a region by the graphs of two functions f(x) and g(x), and by two lines x = a and x = b.

Although you could find this as the difference of the areas of two regions , calculated as


it is often simpler to find this as a single integral





 Ex :

Show that the graphs of f(x) = x3 - x2 - 6x + 8 and g(x) = x3 + 2x2 - 1 intesect at two points , and find the area enclosed between them.

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