Question

For each pair of functions f and g and each interval [a, b] in Exercises 38–40, use at least eight rectangles to approximate the area of the region between the graphs of f and g on the interval [a, b]. Your work should include graphs of f and g together with the rectangles that you used.

$f\left(x\right)=\sin x,g\left(x\right)=\cos x,[0,\pi ]$

Short answer

${\int }_0^{\pi } \left|f\right(x)-g(x\left)\right|dx=2.82$

Step-by-step solution

Step 1. Given information

$f\left(x\right)=\sin x,g\left(x\right)=\cos x,[0,\pi ]$

Step 2. Calculating the area

Below is the graph of both functions,

Area between the graphs,

$\begin{array}{r}\\ ={\int }_0^{\frac{\pi }{8}} (\cos x-\sin x)dx+{\int }_{\frac{\pi }{8}}^{\frac{\pi }{4}} (\cos x-\sin x)dx+{\int }_{\frac{\pi }{4}}^{\frac{3\pi }{8}} (\sin x-\cos x)dx+{\int }_{\frac{3\pi }{8}}^{\frac{\pi }{2}} (\sin x-\cos x)dx\\ +{\int }_{\frac{\pi }{2}}^{\frac{5\pi }{8}} (\sin x-\cos x)dx+{\int }_{\frac{\pi }{2}}^{\frac{3\pi }{4}} (\sin x-\cos x)dx+{\int }_{\frac{3\pi }{4}}^{\frac{7\pi }{8}} (\sin x-\cos x)dx+{\int }_{\frac{7\pi }{8}}^{\pi } (\sin x-\cos x)dx\\ =2\left(\frac{\sqrt{2}}{2}\right)+2\left(\frac{\sqrt{2}}{2}\right)\\ =2\sqrt{2}\\ \approx 2.828\end{array}$