Areas Under Curves
The area under a curve of a function is the integral of that function.
We have met this before in Key Point 3.4 as a path integral and more specifically in 1D with a variable force as

And we have also met it previously in kinematics

Here are some examples for you to practise. In each case, sketch the curve (... you practised that a few weeks ago!) and evaluate the area under the curve between the specified limits. Then do the integration to confirm you answer.
, where F(x) is the restoring force generated by a spring, such that F(x)=-kx where k is the spring constant and x the extension of the spring.