Mathematics

Mathematics

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. a. (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. b. (ii) Using the result of part (i), determine ∫82s(x) dx\displaystyle \int_{8}^{2} s(x)\,dx and explain the change of sign. c. (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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