Lesson Notes By Weeks and Term v4 - SHS 3
APPLICATION OF CALCULUS
Download the Lessonotes Mobile Ghana app for faster lesson access on Android and iPhone.
APPLICATION OF CALCULUS
Download the Lessonotes Mobile Ghana app for faster lesson access on Android and iPhone.
Subject: Additional Mathematics
Class: SHS 3
Term: 1st Term
Week: 17
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
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
This lesson introduces a powerful application of integral calculus: calculating the volume of three-dimensional objects. We will learn how to take a two-dimensional area, rotate it around an axis, and find the volume of the resulting solid of revolution. This method has practical applications in many fields, from traditional crafts like pottery to modern engineering and design. Imagine a potter shaping a lump of clay on a spinning wheel; the final shape of the pot is a solid of revolution. By the end of this lesson, you will be able to mathematically determine the capacity of such objects.
A. 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 (the axis of revolution) that lies in the same plane. Visualisation: Imagine you have a flat shape, like a semi-circle cut out of paper. If you tape the straight edge to a pencil and spin the pencil quickly, the semi-circle will trace out a sphere in the air. That sphere is a solid of revolution.
In this lesson, we will focus on rotating the area under a curve `y = f(x)` around the x-axis or the y-axis. B. The Method of Disks
To find the volume of these complex shapes, we use a clever strategy from calculus. We slice the solid into an infinite number of very thin circular disks (like slicing a carrot or a yam). Volume of a Single Disk: Each disk is essentially a very short cylinder. The volume of a cylinder is `V = πr²h`. `r` is the radius of the disk. `h` is the height (or thickness) of the disk. Connecting to Calculus: If we rotate the curve around the x-axis, the thickness of each disk is a tiny change in x, which we call `dx`. The radius `r` of the disk at any point `x` is simply the height of the curve at that point, which is `y` or `f(x)`. So, the volume of one infinitesimally thin disk (`dV`) is: `dV = π * (radius)² * (thickness)` `dV = π * [f(x)]² * dx` Finding the Total Volume: To get the total volume `V` of the solid, we "add up" (integrate) the volumes of all these tiny disks from our starting point `x = a` to our ending point `x = b`. C. Rotation About the x-axis (Horizontal Axis)