Finding Areas with Integration (Cambridge International IGCSE Further Maths)

Area Under a Curve

What does area under a curve mean?

• The phrase “area under a curve” refers to the area bounded by …

  • ... the x-axis
  • … the graph of y = f(x)
  • … the vertical line x = a
  • … the vertical line x = b

How do I find the area under a curve?

• The value from definite integration is equal to the “area under a curve”

What if I am not told the limits?

• If limits are not provided they will likely be the x-axis intercepts

  • Set y = 0 and solve the equation to find the x-axis intercepts

What happens if the graph is below the x-axis?

• If the area lies underneath the x-axis the value of the integral will be negative
• An area cannot be negative, so take the modulus of the integral

What if the area is made up of more than one section?

• Be careful when one section is above and one section is below the x-axis

  • You will need a separate integral for each section, BUT...
  • ...One section's integral will be negative
  • ...One section's integral will be positive
  • So you’ll need to take the modulus before adding to find the total area

Exam Hint

• Add information to any diagram provided in the question, as well as axes intercepts and values of limits
• Mark and shade the area you’re trying to find, and if no diagram is provided, sketch one!

Worked Example