Area between Two Curves

Question Stem for Question Nos. 15 and 16

Consider the curve $$C_1$$ given by $$y=e^{-x}$$ for $$x\in[0,10\pi]$$, and the curve $$C_2$$ given by $$y=e^{-x}(\sin x+\cos x)$$ for $$x\in[0,10\pi]$$.

Let $$n$$ be the total number of points of intersection of the curves $$C_1$$ and $$C_2$$.

Suppose that $$\alpha_1,\alpha_2,\dots,\alpha_n\in[0,10\pi]$$ are the $$x$$-coordinates of the points of intersection of the curves $$C_1$$ and $$C_2$$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n$$.

Let $$\beta$$ be the area of the region enclosed between the curves $$C_1$$, $$C_2$$, and the lines $$x=\alpha_1$$ and $$x=\alpha_4$$. Then the value of

$$-\dfrac{1}{\pi}\log_e\!\left(\beta-2\,e^{-\frac{\pi}{2}}\right)$$

is ___.