Find the area common to the circle x 2 +y 2 = 4 and the line x = 1 (area of the circular region to the right of x=1).

Find the area common to the circle x 2 +y 2 = 4 and the line x =…

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 x²+y²=4 is 2 times integral from 1 to 2 of sqrt(4-x²) dx, which evaluates using the standard formula integral sqrt(a²-x²)dx = (x/2)sqrt(a²-x²)+(a²/2)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.

Read the full Application of Integrals notes →

Find the area common to the circle x2+y2 = 4 and the line x = 1 (area of the circular region to the right of x=1).

Concept context

More Application of Integrals questions

Find the area common to the circle x²+y² = 4 and the line x = 1 (area of the circular region to the right of x=1).