A liquid flows through a straight horizontal pipe

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

A liquid flows through a straight horizontal pipe. The concentration of a dissolved chemical (in mg/cm) at a distance xx cm from the inlet is modeled by C(x)=x sin⁡(ln⁡(x2+4))x2+4.C(x)=x\,\dfrac{\sin\bigl(\ln(x^{2}+4)\bigr)}{x^{2}+4}. The pipe extends from x=0 cmx=0\text{ cm} to x=2 cmx=2\text{ cm}. (a) Formulate, but do not yet evaluate, an expression that represents the total amount of the chemical (in mg) contained in the section of pipe from x=0x=0 to x=2x=2. (b) Evaluate the definite integral you set up in part (a) to determine the total amount of chemical in that section of pipe. Express your answer in exact form. (c) Interpret your numerical result from part (b) in the context of the problem, including appropriate units.

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