Lesson Notes By Weeks and Term v4 - SHS 3
APPLICATION OF CALCULUS
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APPLICATION OF CALCULUS
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Subject: Additional Mathematics
Class: SHS 3
Term: 1st Term
Week: 16
Grade code: 3.3.2.LI.2
Strand code: 3
Sub-strand code: 2
Content standard code: 3.3.2.CS.1
Indicator code: 3.3.2.LI.2
Theme: CALCULUS
Subtheme: APPLICATION OF CALCULUS
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In our previous lessons, we learned how to find the area under a single curve using definite integration. We saw this as finding the area between the curve and the x-axis. In the real world, however, spaces and shapes are often defined by the intersection or boundary of two or more lines or curves. Imagine a farmer in the Volta Region whose land is bordered by a winding river on one side and a curved road on the other. How can they calculate the exact area of this farm for planning or sale? This lesson provides the mathematical tools to solve such problems. We will extend our knowledge of integration to calculate the area of a region enclosed between two intersecting curves.
A. Recap: Area Under a Single Curve Remember that the definite integral `∫[a, b] f(x) dx` gives us the net area between the curve `y = f(x)` and the x-axis, from `x = a` to `x = b`. B. The Core Concept: Area Between Two Curves
Imagine we have two functions, `y = f(x)` and `y = g(x)`, and we want to find the area of the region between them from `x = a` to `x = b`. Let's assume that over this interval, the graph of `f(x)` is always above the graph of `g(x)`. The area under the upper curve `f(x)` is `A_upper = ∫[a, b] f(x) dx`. The area under the lower curve `g(x)` is `A_lower = ∫[a, b] g(x) dx`.
The area we want is the "yellow" region in the diagram below. We can find it by subtracting the area under the lower curve from the area under the upper curve.
Area between curves = `A_upper - A_lower` `A = ∫[a, b] f(x) dx - ∫[a, b] g(x) dx` `A = ∫[a, b] [f(x) - g(x)] dx`