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The following diagram shows a semicircle centered at O and diameter AB. A rectangle OPQR is drawn such that P lies along [AB] and Q lies on the circular arc.
(diagram not to scale)
It is given that OP=4.
- [2]
Show that ∠AOQ=180∘−tan−1(4x).
It is also given that the arc AQ has length 17.
- [4]
Find the length x of [OR].
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Integration
· Lesson notes
Integrals are a powerful tool that allow us to calculate the area under a curve.
The essence of integration is to draw rectangles under the curve, each with a height matching the function at each point. Adding up the area of these rectangles gives an approximation for the area under the curve. Increasing the number of rectangles makes the approximation more precise.
The real "trick" of integration is imagining what happens when the number of rectangles approaches infinity. In this case, the width each each rectangle approaches zero, and their combined area approaches that of the true region under the curve.
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