Revision: Integration and Kinematics

Let \(f\) and \(g\) be continuous on \([a,b]\). If \(f(x) \geq g(x)\) for all \(x\) in \([a,b]\), the area \(\mathcal{A}\) of the region bounded by the curves \(y=f(x)\) and \(y=g(x)\) and the lines \(x=a\) and \(x=b\) is given by: $$ \mathcal{A} = \int_a^b (f(x)-g(x))\; \mathrm{d}x $$

To find the total geometric area enclosed by two curves \(y=f(x)\) and \(y=g(x)\) (\(f\) and \(g\) continuous, with finitely many intersection points on the interval considered):

1. Find points of intersection. Solve the equation \(f(x)=g(x)\) to find where the curves meet: these points give the bounds \(a\) and \(b\) when they are not given and, when they are, the points inside \([a, b]\) where the upper and lower functions may swap. Choose the consecutive intersection points that bound the region sought.
2. Identify the upper and lower function. On each interval between intersection points, determine which function is on top. Between two consecutive intersection points, \(f - g\) is continuous and does not vanish, so it keeps a constant sign: testing one point is enough (a quick sketch helps).
3. Set up and evaluate the integral(s). On each interval \([c, d]\) between consecutive intersection points or bounds, calculate \(\int_c^d (\text{upper function} - \text{lower function}) \, \mathrm{d}x\), which is non-negative. The total area is the sum of these integrals.

The total distance travelled over \([t_1, t_2]\) is the definite integral of the speed: $$ \text{Distance} = \int_{t_1}^{t_2} |v(t)| \,\mathrm{d}t. $$