Integral of Absolute Value

Rarely Tested
$$\int_a^b |f(x)| \, dx = \int_a^c f(x) \, dx - \int_c^b f(x) \, dx$$
where $$f(c)=0$$ and $$f \geq 0$$ on $$[a,c]$$, $$f \leq 0$$ on $$[c,b]$$.
Question 1

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 ___.

Question 2

Evaluate the value of the definite integral expression defined by:
$$I = 4 \int_{0}^{2} |x^3 - 3x^2 + 2x| \, dx$$

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