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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.

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Furthermore

Formula Booklet:

Area of region enclosed by a curve and y-axis

\( A = \int_{a}^{b} |x| dy \)

Volume of revolution about the x or y-axes

\( V = \int_{a}^{b} \pi y^2 dx \; \text{ or } \; V = \int_{a}^{b} \pi x^2 dy\)

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:

$$V = \pi \int_{a}^{b} (f(x))^2 , dx$$

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:

$$V = \pi \int_{0}^{1} (x)^2 , dx$$

Integrating the above expression, we get:

$$V = \pi \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{\pi}{3}$$

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 \(\frac{\pi}{3}\).

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