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International Baccalaureate Mathematics

Calculus

Syllabus Content

Area of the region enclosed by a curve and the y-axis in a given interval.
Volumes of revolution about the x-axis or y-axis.

Formula Booklet:

Area of region enclosed by a curve and y-axis

A=ba|x|dy

Volume of revolution about the x or y-axes

V=baπy2dx or V=baπx2dy


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Furthermore

The Volume of Revolution about the x-axis is a method used to find the volume of a solid obtained by rotating a region bounded by a function y=f(x), the x-axis, and two vertical lines x=a and x=b about the x-axis.

The formula for finding the volume of the solid is given by:

V=πba(f(x))2,dx

where V represents the volume of the solid.

For example, let us find the volume of the solid obtained by rotating the region bounded by the function y=x and the x-axis about the x-axis from x=0 to x=1:

V=π10(x)2,dx

Integrating the above expression, we get:

V=π[x33]10=π3

Therefore, the volume of the solid obtained by rotating the region bounded by the function y=x and the x-axis about the x-axis from x=0 to x=1 is π3.


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