Which one of the following is equal to the shaded area above?

Question

BlackTom AI · Accepted Answer

Area between two curves on [a,b] equals the integral of (top function − bottom function) dx. When the relative order of f and g changes at an interior point, split the integral at those intersection points and take the positive (top − bottom) on each subinterval.

Relevant concepts:
- Signed area: ∫[a to b] (f − g) dx gives net (signed) area; absolute geometric area is ∫[a to b] |f − g| dx.
- If f(x) ≥ g(x) on an interval, area contribution = ∫(f − g) dx; if f(x) ≤ g(x), area = ∫(g − f) dx.
- Intersections partition the domain into subintervals where the top function is consistent.

Step-by-step reasoning and intermediate algebra:
1. The shaded region runs from x = 0 to x = 4. The curves cross at x = 0 and x = 3, so split [0,4] into [0,3] and [3,4].
2. On [0,3] the graph shows f(x) lies above g(x), so the local area = ∫_0^3 [f(x) − g(x)] dx.
3. On [3,4] g(x) lies above f(x), so the local area = ∫_3^4 [g(x) − f(x)] dx.
4. Total shaded area = ∫_0^3 [f(x) − g(x)] dx + ∫_3^4 [g(x) − f(x)] dx.

Algebraic confirmation:
Area = ∫_0^4 |f − g| dx = ∫_0^3 (f − g) dx + ∫_3^4 −(f − g) dx = ∫_0^3 (f − g) dx + ∫_3^4 (g − f) dx.

Why the correct choice (b) is correct:
Choice (b) exactly matches the expression above: ∫_0^3 [f(x) − g(x)] dx + ∫_3^4 [g(x) − f(x)] dx, which sums positive contributions on each subinterval.

Why the other options are incorrect:
- (a) has the signs reversed on each subinterval, giving negative of the desired area.
- (c) ∫_0^4 [f − g] dx is the signed net area (difference between the two region contributions), not the total geometric area.
- (d) ∫_0^4 [g − f] dx similarly gives the signed area with opposite sign.

Therefore the total shaded area is choice b, confirming the provided answer.

Which one of the following is equal to the shaded area above?单项选择题

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