Question

If the area under the curve \( y=\sqrt{a^2-x^2} \) included between the lines \( x=0 \) and \( x=a \) is 4 sq units. Then the value of \( a \) is

• A. \( \frac{16}{\sqrt{\pi}} \)
• B. \( \frac{4}{\sqrt{\pi}} \)
• C. \( \frac{2}{\sqrt{\pi}} \)
• D. \( -\frac{4}{\sqrt{\pi}} \)

Hint:

The curve \( y=\sqrt{a^2-x^2} \) represents the upper semicircle of radius \(a\)Area from \(0\) to \(a\) is one-fourth of the full circle.

Answer 1

The Correct Option is B

Step 1: Identify the curve.
\[ y=\sqrt{a^2-x^2} \]
This represents the upper semicircle:
\[ x^2+y^2=a^2 \]

Step 2: Understand the required region.
The area between \(x=0\) and \(x=a\) under the curve is one-fourth of the circle.

Step 3: Write area of full circle.
\[ \text{Area of full circle}=\pi a^2 \]

Step 4: Write area of required quarter circle.
\[ \text{Required area}=\frac{\pi a^2}{4} \]

Step 5: Use given condition.
\[ \frac{\pi a^2}{4}=4 \]

Step 6: Solve for \(a\).
\[ \pi a^2=16 \]
\[ a^2=\frac{16}{\pi} \]
\[ a=\frac{4}{\sqrt{\pi}} \]

Step 7: Final conclusion.
Since \(a\) represents radius, it must be positive.
\[ \boxed{\frac{4}{\sqrt{\pi}}} \]