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.3
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.2
Indicator code: 2.3.1.LI.3
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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This lesson introduces the fundamental principles of integral calculus. We have already learned about differentiation, which helps us find the rate of change (like the gradient of a curve). Now, we will explore its opposite: integration. Integration helps us solve two major problems: finding a function when its derivative is known, and calculating the area of irregular shapes bounded by curves. This skill is vital in fields like engineering, economics, and physics, from calculating the area of a piece of farmland by the Pra River to determining the total electricity consumed by a factory over a day.
Part 1: Integration as the Reverse of Differentiation (Anti-derivative)
We know that differentiation finds the gradient function of a given function. For example: If $y = x^3$, then the derivative is $\frac{dy}{dx} = 3x^2$.
Now, let's ask the reverse question: What function, when differentiated, gives us $3x^2$? From our example, we know the answer is $x^3$. This reverse process is called integration or anti-differentiation.
However, consider these other functions: If $y = x^3 + 5$, then $\frac{dy}{dx} = 3x^2$. If $y = x^3 - 100$, then $\frac{dy}{dx} = 3x^2$. If $y = x^3 + C$ (where C is any constant), then $\frac{dy}{dx} = 3x^2$.