Evaluate ∫_{-a}^{a} sqrt(a^2 - x^2) dx. Which value is correct?

Evaluate ∫_{-a}^{a} sqrt(a^2 - x^2) dx. Which value is correct?

• A. (π a^2)/4
• B. (π a^2)/2
• C. (π a^2)
• D. 0

Answer: (B)

The integral computes the area under the curve y = sqrt(a^2 - x^2) from -a to a, which is the upper half of a circle with radius a centered at the origin. Since the full circle has area πa^2, the area of this semicircle is half of that, giving (πa^2)/2. This also makes sense because the integrand is nonnegative on [-a, a], so the result must be a positive fraction of the full circle’s area (not zero), and it is not the quarter or the full circle.

The integral computes the area under the curve y = sqrt(a^2 - x^2) from -a to a, which is the upper half of a circle with radius a centered at the origin. Since the full circle has area πa^2, the area of this semicircle is half of that, giving (πa^2)/2. This also makes sense because the integrand is nonnegative on [-a, a], so the result must be a positive fraction of the full circle’s area (not zero), and it is not the quarter or the full circle.