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Finding Areas with Integration

Exam code: 4037
Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Area Under a Curvevideo

Area Under a Curve

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

Example of area under a curve

How do I find the area under a curve?

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

    Definite integration gives the area under a curve
    Example of finding 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

Example of finding the limits of integration

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

    Example of the area being below the x-axis
Another example of the area being below the x-axis

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

Splitting the area into two areas
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