Question: A microwave oven heats substances by aligning the magnetic dipoles in molecules and vibrating them. The force Y that creates the vibration is a function of the angle t, in radians, between the dipoles and the incident microwaves, with Y(0) = 1, as shown in the graph. (A negative value of Y means that the molecules are less excited than in their normal state). The change in excitation that takes place as the angle sweeps from t = \(\alpha \) to t = \(\beta \) is given by the definite integral of Y over the interval [ \(\alpha ,\beta \) ].

a) Suppose \(Y\left( t \right)={{e}^{t}}\left( \cos t+\sin t \right)\). Find \(Y\left( -\frac{5\pi }{4} \right),\,Y\left( -\frac{\pi }{4} \right)\text{ and }Y\left( \frac{3\pi }{4} \right)\).

b) The function \(U\left( t \right)=5+{{e}^{t}}\sin t\) is the magnetic potential function. Show that \(U'\left( t \right)=Y\left( t \right)\).

c) Use the Fundamental Theorem of Calculus to compute the change in excitation as the angle sweeps from t = 0 to t = \(\frac{3\pi }{4}\)

d) Compute the change in excitation as the angle sweeps from t = \(-\frac{5\pi }{4}\) to t = \(-\frac{\pi }{4}\).

e) Compute the change in excitation over one full cycle \(\left[ -\frac{5\pi }{4},\frac{3\pi }{4} \right]\)