Lesson Notes By Weeks and Term v4 - SHS 2
PRINCIPLES OF CALCULUS
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PRINCIPLES OF CALCULUS
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 14
Grade code: 2.3.1.LI.2
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.2
Indicator code: 2.3.1.LI.2
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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In our everyday lives, we can easily find the area of regular shapes like squares, rectangles, and triangles. But what if we need to find the area of an irregular shape, like a farmer's plot of land next to a winding river, or the area under a graph that shows the speed of a tro-tro over time? This is where the principles of calculus become very powerful. Today, we will learn a fundamental technique to approximate the area under a curve by dividing it into small, manageable rectangles. This method is the visual foundation for a major concept in calculus called integration.
The Core Problem: Area of Irregular Shapes
Imagine we want to find the area of the shaded region below the curve `y = f(x)` from `x = a` to `x = b`.
This shape is not a rectangle or a triangle, so we cannot use a simple formula. The core idea of calculus is to approximate this complex shape with many simple shapes that we *do* know how to calculate – rectangles.
Key Terminology: Interval `[a, b]`: This is the range along the x-axis for which we want to find the area. `a` is the starting point (lower limit) and `b` is the ending point (upper limit). Partitioning: This means to divide or "chop up" the interval `[a, b]` into smaller pieces. Sub-intervals: These are the smaller pieces created after partitioning the main interval. If we partition the interval into `n` pieces, we have `n` sub-intervals. Number of Sub-intervals (`n`): This tells us how many rectangles we will use for our approximation. The more rectangles we use, the better our approximation will be. Step Size (`h` or `Δx`): This is the width of each small sub-interval (and therefore, the width of each rectangle). It is constant for all our rectangles. Formula: `h = (b - a) / n` Where: `b` = upper limit of the interval `a` = lower limit of the interval `n` = number of sub-intervals (rectangles)