Question

The area of the region (in sq.units) bounded by the curves \(x^2 + y^2 = 16\) and \(x^2 + y^2 = 6x\) is?

• A. \(4\pi + 4\sqrt{3}\)
• B. \(\frac{2}{3} \left( 4\pi + \sqrt{3} \right)\)
• C. \(\frac{4}{3} \left( 4\pi + \sqrt{3} \right)\)
• D. \(\frac{4\pi + \sqrt{3}}{3}\)

Hint:

To find the area between two curves, subtract the area of one curve from the area of the other, and integrate accordingly.

Answer 1

The Correct Option is C

The area between the curves is found by integrating the difference between the two functions. We start by finding the equations of the curves. 1. The first equation \(x^2 + y^2 = 16\) represents a circle with radius 4 centered at the origin. 2. The second equation \(x^2 + y^2 = 6x\) is a circle shifted to the right with center \((3, 0)\) and radius 3. We then compute the area enclosed by the two curves by finding the area of intersection and subtracting it from the total area of the first circle. After calculation, the area between the two curves is given by: \[ \frac{4}{3} \left(4\pi + \sqrt{3} \right). \]