Answer: (4pi/3) - sqrt(3).
- A (4pi/3) - sqrt(3)
- B (2pi/3) - sqrt(3)/2
- C (8pi/3) - 2sqrt(3)
- D pi - sqrt(3)
Correct answer: A. (4pi/3) - sqrt(3)
Explanation: Area to the right of x=1 inside x<sup>2</sup>+y<sup>2</sup>=4 is 2 times integral from 1 to 2 of sqrt(4-x<sup>2</sup>) dx, which evaluates using the standard formula integral sqrt(a<sup>2</sup>-x<sup>2</sup>)dx = (x/2)sqrt(a<sup>2</sup>-x<sup>2</sup>)+(a<sup>2</sup>/2)sin-1(x/a), giving total area = (4pi/3) - sqrt(3).
The definite integral ∫ₐᵇf(x)dx computes the exact area of the shaded region bounded by the curve, the x-axis, and the vertical lines x=a and x=b - the same idea behind the Riemann sum, but evaluated exactly rather than approximated by rectangles.
Concept context
Using definite integrals to compute the area under a curve, the area between two curves, and the areas enclosed by standard curves like circles, parabolas, and ellipses.