Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Explain definite integrals and area under a curve.
• Compare the effect of using more subintervals to approximate the area.
• Calculate the area under a curve using rectangles.
• Apply these principles to real-life problems.

Lesson summary

This lesson introduces the fundamental concept of finding the area of irregular shapes bounded by curves. In real life, not everything is a perfect square or circle. Think about the shape of a plot of land by the Volta River, the profile of a hill like Afadjato, or the changing speed of a tro-tro over time. Calculus provides a powerful method to calculate these kinds of areas precisely. We will start by approximating these areas using a shape we know very well: the rectangle. By using more and more rectangles, we will discover how to make our approximation incredibly accurate, leading us to the idea of the definite integral.

Lesson notes

The Core Problem: Area of Irregular Shapes We have formulas to find the area of regular shapes like rectangles (Area = length × width) and triangles (Area = ½ × base × height). But what if we need to find the area of the region under the curve of a function, say `f(x) = x²` between `x=0` and `x=3`?


This shape is not a rectangle or a triangle. The top boundary is curved. The big idea in calculus is to approximate this area by slicing it into many thin vertical strips and treating each strip as a rectangle. The Method: Approximation with Rectangles (Riemann Sums) Interval [a, b]: This is the range on the x-axis we are interested in. In our example, the interval is `[0, 3]`. Number of Subintervals (n): This is the number of rectangles we decide to slice the area into. The more rectangles, the better the approximation. Width of each Subinterval (Δx): This is the base of each rectangle. To make it simple, we make all rectangles have the same width. Formula: `Δx = (b - a) / n` `b` is the upper limit of the interval (e.g., 3). `a` is the lower limit of the interval (e.g., 0). `n` is the number of rectangles. Height of each Rectangle: The height of each rectangle is determined by the value of the function `f(x)` at a certain point within that subinterval. A common and simple method is to use the right-hand endpoint of each subinterval to determine the height.

Evaluation guide

• Compare and judge the effect of reduction in step size and increase in the number of
• subintervals in an interval for a given function on the area under a curve.
• Talk for Learning, Think -pair- share, Experiential Learning and Group
• conceptual