5.5 & 5.8 Integration and Trapezoidal rule

In this section of the course you will learn about an area of calculus called integration. This an be considered in multiple ways, both as the inverse of differentiation (anti-differentiation) and in finding areas under curves. We will also use the trapezoidal rule to estimate the area under a curve. 

Key Concepts

In this unit you should learn to...

  • Integrate polynomial functions. 
  • Find an original function through integration (anti-differentiation) with a boundary condition. 
  • Evaluate definite integrals on the GDC and hence find the area between a curve, the x-axis and an upper and lower bound. 
  • Estimate areas under curves using the trapezoidal rule. 

Essentials

Keep track of your progress on this page and practice the exam questions on this Integration and Trapezoidal Rule activity sheet.

 

1. Introducing Integration as Anti-differentiation. 

This is an introduction to integration with worked examples.

2. Anti-differentiation with a Boundary Condition

This video works through a number of examples of finding an original function. There will be specific attention to questions which have a different notation and style, but ask exactly the same thing. 

3. Definite Integration

What is the different between integration you've done previously and definite integration? How is integration connected to area?

 4. The Trapezoidal Rule 

This video provides everything you need to know about estimating the area under a curve using the trapezoidal rule. 

Summary

These slides summarise the essential understanding and skills in this topic. 

Test Yourself

Click on the hidden box icon to start the quiz.

Exam Style Questions

Question 1

Video Solution

Question 2

Video Solution

Question 3

Video solution

Just for Fun

Have you ever wondered about the connection between the area and circumference of a circle?

If you integrate the circumference you obtain the area:

\(\int_{0}^{r}2\pi r\ dr\ =\ \pi r^{2}\)

Why?

The integral sign actually stands for "infinite sum". That's the reason why it looks like an elongated "S" shape. So the the equation above means the following:

If you draw an infinite number of smaller circles inside a circle, then sum all of the infinite circumferences, then those infinite lines added up give you an area (the more lengths you add, the closer you'll get to the area).      

MY PROGRESS

How much of 5.5 & 5.8 Integration and Trapezoidal rule have you understood?