Integration – Area under a graph

Integration can be used to find the area bounded by a curve y = f(x), the x-axis and the lines x=a and x=b by using the following method.

Area under the graph

However, you must be very careful in the way you use this as the following examples will show.

Area bounded by a curve when it is above the x-axis

In this tutorial I show you how to find the area bounded by a curve when it is above the x-axis.

Area under the graph

Find the area:

  1. \text{Find the area bound by the curve } {y = x^2 + 1}\text{, } \\ \text{the lines } x = -1 \text{, } x = 3 \text{ and the x-axis}
How to find the area bounded by a curve above the x-axis (tutorial 1) : ExamSolutions - youtube Video

Finding the area below the x-axis

In this tutorial I show you how to find the area bounded by a curve when it is below the x-axis.

Area under the graph

Find the area:

  1. \text{Find the area bounded by the curve } {y = x(x + 2)} \text{ and the x-axis.}
Area bounded by a curve below the x-axis (tutorial 2) : ExamSolutions - youtube Video

Finding an area above and below the x-axis

In this tutorial I show you how to find the area bounded by a curve when it is partly below the x-axis and partly above the x-axis using the following example.

Area under the graph

Find the area:

  1. \text{Find the area bound by the curve } {y = (x + 1)(x - 2)}\\  \text{and the x-axis and the line x = 3.}
Area bounded by a curve above and below the x-axis (tutorial 3) : ExamSolutions - youtube Video

Finding the area bounded by two curves

In this tutorial I show you several ways of how to find the area bounded by two curves using the following example.
Area under the graph

Find the area:

  1. \text{Find the area bounded by the curve } {y = {x^2}} \text{ and the line y = x}
How to find the area bounded by two curves (tutorial 4) : ExamSolutions - youtube Video