Answer: 2(sqrt(2)-1).
- A 2(sqrt(2)-1)
- B sqrt(2)-1
- C 2sqrt(2)
- D sqrt(2)
Correct answer: A. 2(sqrt(2)-1)
Explanation: They intersect at x=pi/4. Area = integral 0 to pi/4 (cos x - sin x)dx + integral pi/4 to pi/2 (sin x - cos x)dx. Each piece evaluates to (sqrt2-1), so total = 2(sqrt(2)-1).
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.