International Baccalaureate Mathematics

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.

Here are some exam-style questions on this statement:

"A function $f$ is defined as $f(x) = \arcsin(x — 5)+\frac{\pi}{2}$, where $4 \le x \le 6 $." ... more

"The function $ f $ is defined by $ f(y) = \sqrt{y^2 + 10y} $ for $ 0 \leq y \leq 8 $." ... more

"The function $ f $ is defined as $ f(x) = \sqrt{x \cos(x^2)} $, where $ 0 \leq x \leq \sqrt{\frac{\pi}{2}} $." ... more

"A function $ f $ is defined by $ f(x) = \frac{ke^{\frac{x}{2}}}{1 + e^x} $ where $ x \in \mathbb{R} $, $ x \geq 0 $ and $ k \in \mathbb{Z}^+ $." ... more

"Consider the function $f(x) = \sec^{-1}(x) \text{ where } 1 \le x \le \frac{\pi}{2} $" ... more

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Calculus It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis. Isaac Newton is best known for developing calculus. He was born in England in 1642 and is considered one of the most influential scientists and mathematicians in history.

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