Answer to Question #99233 in Calculus for Leo

Question #99233
For the function v = 12 sin 40, calculate the: a) mean b) root mean square (RMS)
Over a range of 0 ≤ Ø ≤ π/4 radians.

(Note: the trigonometric identity cos 2Ø = 1-2sin^2 Ø)
1
Expert's answer
2019-12-04T09:39:42-0500

a)


"v_{av}=12\\frac{4}{\\pi}\\int _0^{\\frac{\\pi}{4}}d\\theta \\sin{4\\theta}=3\\frac{4}{\\pi}\\int _0^{\\pi}dx \\sin{x}"

"v_{av}=-3(\\cos{\\pi}-\\cos{0})\\frac{4}{\\pi}=\\frac{24}{\\pi}"

b)


"v_{rms}^2=12^2\\frac{4}{\\pi}\\int _0^{\\frac{\\pi}{4}}d\\theta \\sin^2{4\\theta}=12^2\\frac{2}{\\pi}\\int _0^{\\frac{\\pi}{4}}d\\theta (1-\\cos{8\\theta})"

"\\int _0^{\\frac{\\pi}{4}}d\\theta (1-\\cos{8\\theta})=\\frac{\\pi}{4}-\\frac{1}{8}\\int _0^{2\\pi}dy \\sin{y}=\\frac{\\pi}{4}"

Thus,


"v_{rms}^2=12^2\\frac{4}{\\pi}\\int _0^{\\frac{\\pi}{4}}d\\theta \\sin^2{4\\theta}=12^2\\frac{2}{\\pi}\\frac{\\pi}{4}"

"v_{rms}=\\frac{12}{\\sqrt{2}}"


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