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

Question

BlackTom AI · Accepted Answer

Area between two curves equals the integral of (top function − bottom function) over each interval where their vertical order is fixed.

1) Relevant knowledge points:
- The area between curves y = f(x) and y = g(x) on [a,b] is ∫_a^b |f(x) − g(x)| dx.
- If the curves intersect inside [a,b], split the integral at the intersection points and on each subinterval use (top − bottom) so the integrand is nonnegative.

2) Step-by-step reasoning:
- From the graph the curves meet at x = 0 and again at x = 3. For 0 ≤ x ≤ 3 the curve labeled f(x) lies above g(x), so the integrand should be f(x) − g(x) on [0,3]. For 3 ≤ x ≤ 4 the curve labeled g(x) is above f(x), so the integrand should be g(x) − f(x) on [3,4].
- Therefore the shaded area equals the sum of the areas on the two subintervals with the top-minus-bottom integrands.

4) Why the correct answer is correct:
- This expression guarantees a nonnegative integrand on each subinterval and exactly measures the vertical distance between the top and bottom curves over the entire shaded region.

5) Why the other choices are incorrect:
- (a) ∫_0^4 [g(x) − f(x)] dx assumes g is above f on the whole interval; that is false on [0,3], so this would give negative contributions on [0,3].
- (b) ∫_0^3 [g − f] dx + ∫_3^4 [f − g] dx is the reverse signs on both subintervals (the negative of the correct expression), so it yields the negative of the desired area.
- (d) ∫_0^4 [f − g] dx assumes f is above g on the entire interval; this is false on [3,4] and would subtract the area on [3,4] instead of adding it.

6) Conclusion:
Thus the shaded area equals ∫_0^3 [f(x) − g(x)] dx + ∫_3^4 [g(x) − f(x)] dx, which matches answer choice C.

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

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