12.7 Applications of Integration

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

Negative areas

•  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