Interpreting a Definite Integral

Determine if each definite integral can be interpreted as an area. If it can, describe the area; if it cannot, explain why.

1. \(\int_{0}^{3\pi /4}\cos x\,dx\)
2. \(\int_{2}^{10}\left \vert x-4\right\vert dx\)

Solution (a) See Figure 19. Since \(\cos x\lt0\) on the interval \(\left(\dfrac{\pi }{2},\dfrac{3\pi}{4}\right],\) the integral \(\int_{0}^{3\pi /4}\cos x\,dx\) cannot be interpreted as area.

(b) See Figure 20. Since \(\vert x-4\vert \geq 0\) on the interval \([2,10] \), the integral \(\int_{2}^{10}\vert x-4\vert \,dx\) can be interpreted as the area enclosed by the graph of \(y=\vert x-4\vert \), the \(x\)-axis, and the lines \(x=2\) and \(x=10.\)

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