Question

Let $A_1$ be the area of the region bounded by the curves $y=\sin \left(x\right)$, $y=\cos \left(x\right)$, and $y$-axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by the curves $y=\sin \left(x\right)$, $y=\cos \left(x\right)$, the $x$-axis and $x=\frac{\mathrm{\pi }}{2}$ in the first quadrant. Then,

• A. $A_1=A_2\text{and}{A}_1+A_2=\sqrt{2}$
• B. $A_1:A_2=1:2\text{and}{A}_1+A_2=1$
• C. $2A_1=A_2\text{and}{A}_1+A_2=1+\sqrt{2}$
• D. $A_1:A_2=1:\sqrt{2}\text{and}{A}_1+A_2=1$

Answer 1

The correct option is D

$A_1:A_2=1:\sqrt{2}\text{and}{A}_1+A_2=1$

Determine relation between $A_1\text{and}{A}_2$

Step 1: Calculate the value of $A_1$

Since $A_1$ is area bounded by the curves $y=\sin \left(x\right)$, $y=\cos \left(x\right)$, $y$-axis in the first quadrant

we have,

$$\begin{gathered} A_1={\int }_0^{\frac{\mathrm{\pi }}{4}}\left(\cos \left(x\right)-\sin \left(x\right)\right)dx \\ ={\left|\left[\sin \left(x\right)+\cos \left(x\right)\right]\right|}_0^{\frac{\mathrm{\pi }}{4}} \\ =\left[\sin \left(\frac{\mathrm{\pi }}{4}\right)+\cos \left(\frac{\mathrm{\pi }}{4}\right)-\sin \left(0\right)-\cos \left(0\right)\right] \\ =\left[\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-0-1\right] \\ =\sqrt{2}-1 \end{gathered}$$

Step 2: Calculate the value of $A_2$

Since $A_2$ is area bounded by the curves $y=\sin \left(x\right)$, $y=\cos \left(x\right)$, the $x$-axis and $x=\frac{\mathrm{\pi }}{2}$ in the first quadrant, we have,

$$\begin{gathered} A_2={\int }_0^{\frac{\mathrm{\pi }}{2}}\cos \left(x\right)dx-{\int }_0^{\frac{\mathrm{\pi }}{4}}\left(\cos \left(x\right)-\sin \left(x\right)\right)dx \\ ={\left[\sin \left(x\right)\right]}_0^{\frac{\mathrm{\pi }}{2}}-\left(\sqrt{2}-1\right) \\ =\left[\sin \left(\frac{\mathrm{\pi }}{2}\right)-\sin \left(0\right)\right]-\left(\sqrt{2}-1\right) \\ =1-\left(\sqrt{2}-1\right) \\ =2-\sqrt{2} \end{gathered}$$

Step 3: Determine the relation between the two areas

,$$\begin{gathered} \frac{A_1}{A_2}=\frac{\sqrt{2}-1}{2-\sqrt{2}} \\ =\frac{\sqrt{2}-1}{\sqrt{2}\left(\sqrt{2}-1\right)} \\ =\frac{1}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} A_1+A_2=\sqrt{2}-1+\left(1-\sqrt{2}+1\right) \\ =1 \end{gathered}$$

Hence, option (D) is the correct answer.