A function is continuous on and satisfies _ 2 ^ 5 s(x) ,dx = 6 and _ 5 ^ 8 s(x) ,dx = -4

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A function ss is continuous on [2,8][2,8] and satisfies ∫25s(x) dx=6and∫58s(x) dx=−4.\int_{2}^{5} s(x)\,dx = 6 \quad\text{and}\quad \int_{5}^{8} s(x)\,dx = -4. Answer the following, giving suitable explanations. (i) Show that ∫28s(x) dx=2\displaystyle \int_{2}^{8} s(x)\,dx = 2. Give a brief justification based on areas under the curve. (ii) Using the result of part (i), determine ∫82s(x) dx\displaystyle \int_{8}^{2} s(x)\,dx and explain the change of sign. (iii) If p(x)=3s(x)p(x)=3s(x), find ∫25p(x) dx\displaystyle \int_{2}^{5} p(x)\,dx without direct integration. State the principle used.

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