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: 18
Grade code: 3.3.2.LI.3
Strand code: 3
Sub-strand code: 2
Content standard code: 3.3.2.CS.1
Indicator code: 3.3.2.LI.3
Theme: CALCULUS
Subtheme: APPLICATION OF CALCULUS
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This lesson explores two powerful applications of integral calculus. First, we will learn how integration can be used as a tool for accumulation, allowing us to calculate a total quantity (like distance travelled or total water in a tank) when we know its rate of change (like velocity or flow rate). This is a fundamental concept in physics, engineering, and economics. Secondly, we will delve into a fascinating geometric application: calculating the exact volume of three-dimensional shapes formed by rotating a two-dimensional curve around an axis.
This topic is divided into two main applications of definite integration. Part 1: Integration as Net Change / Accumulation
The Fundamental Theorem of Calculus tells us that if a function `r(t)` represents the rate of change of a quantity `Q(t)`, then the integral of `r(t)` from time `a` to time `b` gives the total change in the quantity `Q` over that period.
Mathematically: Total Change = `Q(b) - Q(a) = ∫[a, b] r(t) dt`
This means we can find the final amount of a quantity if we know its initial amount and its rate of change: Final Amount = `Q(b) = Q(a) + ∫[a, b] r(t) dt`