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: 20
Grade code: 3.3.2.LI.4
Strand code: 3
Sub-strand code: 2
Content standard code: 3.3.2.CS.1
Indicator code: 3.3.2.LI.4
Theme: CALCULUS
Subtheme: APPLICATION OF CALCULUS
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This lesson explores a powerful application of integral calculus: calculating the volume of three-dimensional objects. Many objects we see and use in Ghana are created by rotating a two-dimensional shape around an axis. Think about a potter shaping a clay pot on a wheel, the shape of a water tank, a Fanta bottle, or even the dome on a traditional building. These are all "solids of revolution." By using calculus, we can move beyond simple formulas for cylinders and cones and find the exact volume of these more complex, curved shapes. This skill is vital in fields like engineering, architecture, design, and manufacturing.
What is a Solid of Revolution?
A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional plane curve around a straight line (called the axis of revolution). Visualisation (Think-Pair-Share Activity): Imagine a flat, rectangular piece of paper (like a flag on a pole). If you rotate this rectangle around the edge attached to the pole (the axis), what 3D shape do you get? (Answer: A cylinder). Now, imagine a right-angled triangle. If you rotate it around one of its shorter sides, what shape do you get? (Answer: A cone).
The shapes we will be dealing with are formed by rotating a region under a curve. The Disk Method: Revolution about the x-axis
This is the fundamental method we will use. Let's break it down logically. The Region: We start with a continuous function `y = f(x)` on an interval `[a, b]`. We are interested in the area bounded by this curve, the x-axis, and the vertical lines `x = a` and `x = b`. Slicing the Region: Imagine slicing this area into many, very thin vertical rectangles. Let the width of each rectangle be a tiny change in x, which we call `Δx`. The height of a rectangle at any point `x` is `y = f(x)`. Revolving a Single Slice: Now, take one of these thin rectangles and rotate it 360° around the x-axis. It will form a thin, flat cylinder, which we call a disk (like a coin). The radius of this disk is the height of the rectangle, `r = y = f(x)`. The thickness (or height) of this disk is the width of the rectangle, `h = Δx`. Volume of One Disk: We know the volume of a cylinder is `V = πr²h`. So, the volume of our single disk (`ΔV`) is: `ΔV ≈ π * [f(x)]² * Δx` Summing the Disks: To find the total volume of the solid, we need to "add up" the volumes of all these infinitesimally thin disks from `x = a` to `x = b`. In calculus, this "summing up" is done by integration.