W5.2 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

$$ W = \int^{\rm finish}_{\rm start} F(x) dx $$

And we have also met it previously in kinematics

$$ v = \frac{ds}{dt} \Rightarrow s = \int v(t) dt $$

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.

  1. $\int^{\pi}_{0} \sin (x) dx $
  2. $\int^{\pi}_{0} \cos (x) dx $
  3. $\int^{x_1}_{x_0} F(x) dx $, 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.

Solution

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Solution

  1. $\int^{\pi}_{0} \sin (x) dx = -\left[ \cos(x) \right]^{\pi}_{0} = -\left[ (-1)-(1) \right] = 2$
  2. $\int^{\pi}_{0} \cos (x) dx = \left[ \sin(x) \right]^{\pi}_{0} = \left[ 0 - 0 \right] = 0$
  3. $\int F(x) dx = \int (-kx) dx = - 1/2k(x^2_1 - x^2_0) $