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Revision:OCR Core 3 - Volumes of revolution
11. Volumes of revolutionSolids of revolutionConsider some line that is drawn, now consider a solid that is formed by allowing this line to rotated by four-right-angles about the x-axis, this is a solid of revolution, whose volume is a volume of revolution. Some of the solids of revolution are shapes which a candidate would known how to calculate the volume of, such as a cone, or cylinder, however others are new solids, and therefore need to be considered in a different manner. Assume that the volume can be expressed as a function of "x", and "y". An increase in "x" causes an increase in "y", and hence there is an increase in "V". There is a cylinder formed whose radius is The change in volume is between the volume of the aforementioned cylinder, and the one before the increase. As the change in "x" tends to zero, so does the change in "y", hence:
Example 1. Calculate the volume of the solid formed when the graph of
Hence:
Volumes of revolution about the y-axisThe same argument as was used for rotation about the x-axis applies to the y-axis, and therefore it is easy to state that:
There is rearrangement of the equation usually to do, as the equation of the curve will be given (in most cases) in the form If there is a problem when rearranging then this cannot be used (for instance if the function is not defined over the specified limit). Rotating regions between curvesSometimes it is necessary to calculate the volume of the solid of rotation of the region between two curves. In Core 2 the topic of the area between two curves was addressed, the method was to subtract one area from the other; with volumes there is no change, and therefore a very similar method can be used.
Often the candidate will be required to supply the limits of integration as the total area enclosed by the curves is simply going to have limits where these curves intersect. |
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![V = \pi \int ^{2} _{0} x^{4} \,dx = \pi \left[ \frac{1}{5} x^{5} \right] ^{2}_{0} = \frac{32 \pi }{5}](http://www.thestudentroom.co.uk/latexrender/pictures/87/87123aa3005f58359c215acc434c69a9.png "V = \pi \int ^{2} _{0} x^{4} \,dx = \pi \left[ \frac{1}{5} x^{5} \right] ^{2}_{0} = \frac{32 \pi }{5}")
![V = \pi \int ^{a} _{b} (f(x))^{2} \,dx - \pi \int ^{a} _{b} (g(x))^{2} \,dx = \pi \int ^{a} _{b} \left[ (f(x))^{2} - (g(x))^{2} \right] \,dx](http://www.thestudentroom.co.uk/latexrender/pictures/05/05e47f61e44a2c54885a6416351a1ca3.png "V = \pi \int ^{a} _{b} (f(x))^{2} \,dx - \pi \int ^{a} _{b} (g(x))^{2} \,dx = \pi \int ^{a} _{b} \left[ (f(x))^{2} - (g(x))^{2} \right] \,dx")
