Answer: Split at the crossing point, take absolute values of each piece, and add.
- A Integrate straight across from a to b and ignore the sign of the result
- B Split at the crossing point, take absolute values of each piece, and add
- C Conclude the total area is always zero since signs cancel
- D Integrate once across [a,b] and multiply the result by 2
Correct answer: B. Split at the crossing point, take absolute values of each piece, and add
Explanation: Where the sign of f(x) changes, you must split the interval at the root, evaluate each piece, take absolute values, and sum them to get the correct total area.
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.