The integral \(\displaystyle\int_{-5}^{5}\sqrt{25-x^{2}}\,dx\) gives the area of a region bounded by a curve and the x-axis. The value of the integral is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{25\pi}{2}\)
The curve \(y = \sqrt{25-x^{2}}\) has \(y \ge 0\), and squaring gives
\(x^{2}+y^{2} = 25\)
so it is the upper half of the circle of radius 5. From x = −5 to x = 5 the region under it is the whole upper half-disc:
\( \displaystyle \int_{-5}^{5}\sqrt{25-x^{2}}\,dx = \frac{1}{2}\times\pi\times5^{2} = \frac{25\pi}{2} \)