Mathematics

Mathematics

Problem

Without using absolute value notation, which of the following definite–integral expressions equals the total area of the region bounded by the graphs of continuous functions ff and gg on [a,d][a,d] if the graphs intersect exactly at x=a<b<c<dx=a<b<c<d and satisfy f(x)≥g(x)f(x)\ge g(x) for a≤x≤ba\le x\le b and c≤x≤dc\le x\le d, while f(x)≤g(x)f(x)\le g(x) for b≤x≤cb\le x\le c? a) ∫ab(f(x)−g(x)) dx+∫bd(g(x)−f(x)) dx\displaystyle \int_a^{b}\bigl(f(x)-g(x)\bigr)\,dx+\int_{b}^{d}\bigl(g(x)-f(x)\bigr)\,dx b) ∫ad(f(x)−g(x)) dx\displaystyle \int_a^{d}\bigl(f(x)-g(x)\bigr)\,dx c) ∫ab(f(x)−g(x)) dx+∫bc(g(x)−f(x)) dx+∫cd(f(x)−g(x)) dx\displaystyle \int_a^{b}\bigl(f(x)-g(x)\bigr)\,dx+\int_{b}^{c}\bigl(g(x)-f(x)\bigr)\,dx+\int_{c}^{d}\bigl(f(x)-g(x)\bigr)\,dx d) ∫ab(g(x)−f(x)) dx+∫bc(f(x)−g(x)) dx+∫cd(g(x)−f(x)) dx\displaystyle \int_a^{b}\bigl(g(x)-f(x)\bigr)\,dx+\int_{b}^{c}\bigl(f(x)-g(x)\bigr)\,dx+\int_{c}^{d}\bigl(g(x)-f(x)\bigr)\,dx

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