Medical imaging devices like CT scans work by passing an $X$ -ray beam through part of the body and measuring how much the intensity of the beam attenuates (in other words, is reduced. The amount of attenuation depends on the density and composition of the tissue.
(a) If $A(x)$ is the attenuation rate at position $x$ in the tissue and $L$ is the thickness of the tissue through which the beam passes, what is the total attenuation of the beam during this procedure?
(b) Suppose the maximum and minimum attenuation rates in the tissue are $\beta$ and $\alpha,$ respectively. Find upper and lower bounds for the total attenuation of the beam as it passes through the tissue. Justify your answer using the properties of integrals in Section 5.2 .