Two differentiable functions and intersect at several points on the interval

MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

Two differentiable functions hh and kk intersect at several points on the interval [c,d][c,d]. A student claims that ∫cd∣h(x)−k(x)∣ dx\int_{c}^{d}\lvert h(x)-k(x)\rvert\,dx represents the total area of the region enclosed by the two curves without requiring the student to find those intersection points. Which of the following statements best justifies the student’s claim? (a) Because one may first compute ∫cd(h(x)−k(x)) dx\displaystyle\int_{c}^{d}\bigl(h(x)-k(x)\bigr)\,dx and then take the absolute value of that single number, yielding the total area between the curves. (b) Because the absolute value forces the integrand to be non-negative, any portion where h(x)<k(x)h(x)<k(x) contributes positively to the integral instead of canceling out. (c) Because ∣h(x)−k(x)∣=h(x)−k(x)\lvert h(x)-k(x)\rvert=h(x)-k(x) on the entire interval [c,d][c,d], the integrand is always non-negative. (d) Because definite integrals are automatically non-negative, the sign of h(x)−k(x)h(x)-k(x) does not affect the value of the integral.

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