Mathematics

Mathematics

Problem

Let rr and ss be continuous functions that intersect only at x=2x=2 on the interval [0,4][0,4]. Assume r(x)>s(x)r(x)>s(x) for 0≤x<20\le x<2 and r(x)<s(x)r(x)<s(x) for 2<x≤42<x\le4. Which of the following integrals equals the area of the region enclosed by the graphs of rr and ss between x=0x=0 and x=4x=4? a) ∫02(s(x)−r(x)) dx+∫24(r(x)−s(x)) dx\displaystyle \int_{0}^{2}\bigl(s(x)-r(x)\bigr)\,dx+\int_{2}^{4}\bigl(r(x)-s(x)\bigr)\,dx b) ∫02(r(x)−s(x)) dx+∫24(s(x)−r(x)) dx\displaystyle \int_{0}^{2}\bigl(r(x)-s(x)\bigr)\,dx+\int_{2}^{4}\bigl(s(x)-r(x)\bigr)\,dx c) ∫04(r(x)−s(x)) dx\displaystyle \int_{0}^{4}\bigl(r(x)-s(x)\bigr)\,dx d) ∫02(r(x)−s(x)) dx−∫24(r(x)−s(x)) dx\displaystyle \int_{0}^{2}\bigl(r(x)-s(x)\bigr)\,dx-\int_{2}^{4}\bigl(r(x)-s(x)\bigr)\,dx

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