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Step-by-Step Solution
Step 1: Identify the Regions for $A_1$ and $A_2$
• The region corresponding to $A_1$ is bounded by the curves $y = \sin x$, $y = \cos x$, and the $y$-axis (i.e., $x=0$) in the first quadrant, up to the point of intersection of $\sin x$ and $\cos x$.
• The curves $\sin x$ and $\cos x$ intersect at $x = \frac{\pi}{4}$ when $\sin x = \cos x$.
• Therefore, $A_1$ is the area between $x=0$ and $x = \frac{\pi}{4}$, with the top function $y = \cos x$ and the bottom function $y = \sin x$.
Step 2: Calculate $A_1$
$A_1$ can be found by the definite integral of $(\cos x - \sin x)$ from $0$ to $\frac{\pi}{4}$:
$
A_1 = \int_{0}^{\frac{\pi}{4}} \bigl(\cos x - \sin x\bigr)\,dx.
$
Integrate term by term:
$
\int (\cos x)\,dx = \sin x, \quad \int (-\sin x)\,dx = \cos x.
$
Thus,
$
A_1 = \Bigl[\sin x + \cos x\Bigr]_{0}^{\frac{\pi}{4}}
= \Bigl(\sin \tfrac{\pi}{4} + \cos \tfrac{\pi}{4}\Bigr) - \bigl(\sin 0 + \cos 0\bigr).
$
Evaluating:
$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin 0 = 0, \quad \cos 0 = 1.$
Hence,
$
A_1 = \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\right) - (0 + 1)
= \sqrt{2} - 1.
$
Step 3: Understand $A_2$
• The region corresponding to $A_2$ is bounded by the same curves $y = \sin x$ and $y = \cos x$, along with the $x$-axis, from $x = \frac{\pi}{4}$ to $x = \frac{\pi}{2}$ (since the question states it also includes $x=\frac{\pi}{2}$).
• However, an easier way to find $A_2$ is by noting that $A_1 + A_2$ is simply the total area under $y = \cos x$ from $x=0$ to $x=\frac{\pi}{2}$ (because $\cos x$ is above the $x$-axis on this interval, and $\sin x$ complements the area inside).
Step 4: Calculate $A_1 + A_2$
Compute the integral of $\cos x$ from $0$ to $\frac{\pi}{2}$:
$
A_1 + A_2 = \int_{0}^{\frac{\pi}{2}} \cos x\,dx = \Bigl[\sin x\Bigr]_{0}^{\frac{\pi}{2}}
= \sin \tfrac{\pi}{2} - \sin 0
= 1 - 0
= 1.
$
Step 5: Calculate $A_2$
Since $A_1 + A_2 = 1$ and $A_1 = \sqrt{2} - 1$, it follows that:
$
A_2 = 1 - A_1
= 1 - (\sqrt{2} - 1)
= 2 - \sqrt{2}.
$
Step 6: Find the Ratio $A_1 : A_2$
Compute the ratio:
$
\frac{A_1}{A_2} = \frac{\sqrt{2} - 1}{2 - \sqrt{2}}.
$
Notice that $2 - \sqrt{2} = \sqrt{2}\,(\sqrt{2} - 1).$ Therefore,
$
\frac{A_1}{A_2}
= \frac{\sqrt{2} - 1}{\sqrt{2}(\sqrt{2} - 1)}
= \frac{1}{\sqrt{2}}.
$
Hence $A_1 : A_2 = 1 : \sqrt{2}.$
Step 7: Final Answer
• The ratio is $A_1 : A_2 = 1 : \sqrt{2}$.
• The sum is $A_1 + A_2 = 1.$
Illustration
Thus, the correct option is
$A_1 : A_2 = 1 : \sqrt{2}$ and $A_1 + A_2 = 1.$
