Find the total area between the curve f(x)=cosx and the x-axis on the interval[0,2pi].
Here, we have to be very careful. The question asks for the total area between the function and the x-axis but if we simply take the integral,
notice that we get an answer of 0. This is because the integral counts area above the x-axis as positive and that below the x-axis as negative. This problem contains an equal amount of area above and below the graph. So, we need to split this up into a few integrals
Notice the middle integral has a negative sign out in front. This is because the graph is beneath the x-axis on that interval. Since the integral counts it as negative area, a negative sign in front makes it positive. The two integrals without negative signs in front represent regions where the function is above the x-axis. So, the area is
Thus the total area between the curve and the x-axis on the interval is 4 square units.