Let be defined on

MATHEMATICS · Subjective · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let h(x)=sin⁡x+xh(x)=\sin x + x be defined on [0,π][0,\pi]. (i) Calculate the average value of hh on the sub-interval [0,π2][0,\tfrac{\pi}{2}]. (ii) Calculate the average value of hh on the sub-interval [π2,π][\tfrac{\pi}{2},\pi]. (iii) Without performing any further integration, identify which of the two averages found in parts (i) and (ii) is larger and justify your answer briefly. (iv) Determine the ratio of the larger average to the smaller one, writing your answer in simplest form.

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