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: 15
Grade code: 2.3.1.LI.4
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.2
Indicator code: 2.3.1.LI.4
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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Welcome, future engineers, economists, and scientists! Today, we are exploring a foundational idea in calculus: finding the area of irregular shapes. Imagine you are a surveyor trying to find the area of a piece of farmland bordered by a straight road on one side and a winding river on the other. How would you do it? Standard formulas like `length × width` won't work because of the curved boundary. Calculus provides a powerful method to solve this problem by approximating the area with simple shapes we know (rectangles) and then refining that approximation to get an exact value.
Concept 1: The Problem of "Area Under a Curve"
The "area under a curve" is the area of the region bounded by the graph of a function `f(x)`, the x-axis, and two vertical lines `x = a` and `x = b`. Function: `f(x)` (the curved top boundary) Interval: `[a, b]` (the left and right boundaries on the x-axis)
For a shape with straight sides, like a square or a trapezium, finding the area is easy. But for a curve, we need a new strategy. Concept 2: The Strategy - Approximation with Rectangles (Riemann Sums)
The core idea is to slice the area into thin vertical strips and treat each strip as a rectangle. We can easily find the area of a rectangle (`width × height`) and then add up the areas of all the rectangles.