107 449
Assignments Done
95.9%
Successfully Done
In May 2023

# Answer to Question #213360 in Civil and Environmental Engineering for Elisha

Question #213360

Find the Fourier sine and Fourier cosine integral for the following function

f(x) = (x2, 0 < x ≤ 10

0, x > 10.

1
2021-07-05T04:54:31-0400

"\\displaystyle\n(i)\\\\\n\n\\textsf{Fourier sine transform of}\\,f(x) \\\\\n\\begin{aligned}\nf_s(\\omega) &= \\sqrt{\\frac{2}{\\pi}}\\int_{-\\infty}^{\\infty} f(x) \\sin(\\omega x) \\mathrm{d}x\\\\\n&=\\sqrt{\\frac{2}{\\pi}}\\int_0^{10} x^2 \\sin(\\omega x) \\mathrm{d}x + \\sqrt{\\frac{2}{\\pi}}\\int_{10}^{\\infty} 0 \\times \\sin(\\omega x) \\mathrm{d}x\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\int_0^{10} x^2 \\mathrm{d}(-\\cos(\\omega x))\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(-x^2 \\cos(\\omega x)\\vert_0^{10} + 2\\int_0^{10} x\\cos(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(-100 \\cos(100\\omega) + \\frac{2}{\\omega}\\int_0^{10} x\\mathrm{d}(\\sin(\\omega x))\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(-100 \\cos(100 \\omega) + \\frac{2}{\\omega} x\\sin(\\omega x)\\vert_0^{10} - \\frac{2}{\\omega}\\int_0^{10} \\sin(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(-100 \\cos(100 \\omega) + \\frac{2}{\\omega}x\\sin(\\omega x)\\vert_0^{10} + \\frac{2}{\\omega^2}\\cos(\\omega x)\\vert_0^{10}\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(-100 \\cos(100 \\omega) + \\frac{20}{\\omega}\\sin(20\\omega) + \\frac{2}{\\omega^2}(\\cos(10 \\omega) - 1)\\right)\n\\end{aligned} \\\\\n\n\\textsf{Fourier cosine transform of}\\,f(x)\\\\\n\n\\begin{aligned}\nf_c(\\omega) &= \\sqrt{\\frac{2}{\\pi}}\\int_{-\\infty}^{\\infty} f(x) \\cos(\\omega x) \\mathrm{d}x\\\\\n&=\\sqrt{\\frac{2}{\\pi}}\\int_0^{10} x^2 \\cos(\\omega x) \\mathrm{d}x + \\sqrt{\\frac{2}{\\pi}}\\int_{10}^{\\infty} 0 \\times \\cos(\\omega x) \\mathrm{d}x\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\int_0^{10} x^2 \\mathrm{d}(\\sin(\\omega x))\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(x^2 \\sin(\\omega x)\\vert_0^{10} - 2\\int_0^{10} x\\sin(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(100 \\sin(100\\omega) + \\frac{2}{\\omega}\\int_0^{10} x\\mathrm{d}(\\cos(\\omega x))\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(100 \\sin(100 \\omega) +\\frac{2}{\\omega} x\\cos(\\omega x)\\vert_0^{10} - \\frac{2}{\\omega}\\int_0^{10} \\cos(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(100 \\sin(100 \\omega) +\\frac{2}{\\omega}x\\cos(\\omega x)\\vert_0^{10} - \\frac{2}{\\omega^2}\\sin(\\omega x)\\vert_0^{10}\\right)\\\\\n&= \\sqrt{\\frac{2}{\\pi \\omega^2}}\\left(100 \\sin(100 \\omega) + \\frac{20}{\\omega}\\cos(20\\omega) - \\frac{2}{\\omega^2}\\sin(10 \\omega)\\right)\n\\end{aligned} \\\\\n\n\n\n(ii)\\\\\n\\textsf{Fourier sine transform of}\\,f(x) \\\\\n\n\\begin{aligned}\nf_c(\\omega) &= \\sqrt{\\frac{2}{\\pi}}\\int_{-\\infty}^{\\infty} f(x) \\cos(\\omega x) \\mathrm{d}x\\\\\n&=\\sqrt{\\frac{2}{\\pi}}\\int_0^{1} x\\cos(\\omega x) \\mathrm{d}x + \\sqrt{\\frac{2}{\\pi}}\\int_1^{2} (x + 1)\\cos(\\omega x) \\mathrm{d}x \\\\&+ \\sqrt{\\frac{2}{\\pi}}\\int_{2}^{\\infty} 0 \\times \\cos(\\omega x) \\mathrm{d}x \\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\int_0^{1} x\\mathrm{d}(\\sin(\\omega x)) + \\int_1^{2} (x + 1)\\mathrm{d}(\\sin(\\omega x))\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(x\\sin(\\omega x)\\vert_0^{1} - \\int_0^{1} \\sin(\\omega x) \\mathrm{d}x + (x + 1)\\sin(\\omega x)\\vert_1^{2}\\right.\\\\&\\left. - \\int_1^{2} \\sin(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\sin(\\omega) + \\frac{\\cos(\\omega x)}{\\omega}\\vert_0^{1} +3\\sin(2\\omega) \\right.\\\\&\\left.- 2\\sin(\\omega) + \\frac{\\cos(\\omega x)}{\\omega}\\vert_1^{2}\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\sin(\\omega) + \\frac{\\cos(\\omega)}{\\omega} - \\frac{1}{\\omega} + 3\\sin(2\\omega) \\right.\\\\&\\left.- 2\\sin(\\omega) + \\frac{\\cos(2\\omega)}{\\omega} - \\cos(\\omega)\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(-\\sin(\\omega) + 3\\sin(2\\omega) + \\frac{\\cos(2\\omega)}{\\omega} - \\frac{1}{\\omega}\\right) \n\\end{aligned} \\\\\n\n\\textsf{Fourier cosine transform of}\\,f(x) \\\\\n\n\\begin{aligned}\nf_c(\\omega) &= \\sqrt{\\frac{2}{\\pi}}\\int_{-\\infty}^{\\infty} f(x) \\cos(\\omega x) \\mathrm{d}x\n\\\\&=\\sqrt{\\frac{2}{\\pi}}\\int_0^{1} x\\cos(\\omega x) \\mathrm{d}x + \\sqrt{\\frac{2}{\\pi}}\\int_1^{2} (x + 1)\\cos(\\omega x) \\mathrm{d}x \\\\&+ \\sqrt{\\frac{2}{\\pi}}\\int_{2}^{\\infty} 0 \\times \\cos(\\omega x) \\mathrm{d}x \\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\int_0^{1} x\\mathrm{d}(\\sin(\\omega x)) + \\int_1^{2} (x + 1)\\mathrm{d}(\\sin(\\omega x))\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(x\\sin(\\omega x)\\vert_0^{1} - \\int_0^{1} \\sin(\\omega x) \\mathrm{d}x + \\int (x + 1)\\sin(\\omega x)\\vert_1^{2} \\right.\\\\&\\left.- \\int_1^{2} \\sin(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\sin(\\omega) + \\cos(\\omega x)\\vert_0^{1} +3\\sin(2\\omega) \\right.\\\\&\\left.- 2\\sin(\\omega) + \\cos(\\omega x)\\vert_1^{2}\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\sin(\\omega) + \\cos(\\omega) - 1 + 3\\sin(2\\omega) \\right.\\\\&\\left.- 2\\sin(\\omega) + \\cos(2\\omega) - \\cos(\\omega)\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(-\\sin(\\omega) + 3\\sin(2\\omega) + \\cos(2\\omega) - 1\\right) \n\\end{aligned} \\\\\n\n\n\n\\begin{aligned}\nf_s(\\omega) &= \\sqrt{\\frac{2}{\\pi}}\\int_{-\\infty}^{\\infty} f(x) \\sin(\\omega x) \\mathrm{d}x\\\\\n&=\\sqrt{\\frac{2}{\\pi}}\\int_0^{1} x\\sin(\\omega x) \\mathrm{d}x + \\sqrt{\\frac{2}{\\pi}}\\int_1^{2} (x + 1)\\sin(\\omega x) \\mathrm{d}x \\\\&+ \\sqrt{\\frac{2}{\\pi}}\\int_{2}^{\\infty} 0 \\times \\sin(\\omega x) \\mathrm{d}x \\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\int_0^{1} x\\mathrm{d}(-\\cos(\\omega x)) + \\int_1^{2} (x + 1)\\mathrm{d}(-\\cos(\\omega x))\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(-x\\cos(\\omega x)\\vert_0^{1} + \\int_0^{1} \\cos(\\omega x) \\mathrm{d}x - (x + 1)\\cos(\\omega x)\\vert_1^{2} \\right.\\\\&\\left.+ \\int_1^{2} \\cos(\\omega x) \\mathrm{d}x\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(-\\cos(\\omega) + \\frac{\\sin(\\omega x)}{\\omega}\\vert_0^{1} - 3\\cos(2\\omega) \\right.\\\\&\\left.+2\\cos(\\omega) + \\frac{\\sin(\\omega x)}{\\omega}\\vert_1^{2}\\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(-\\cos(\\omega) + \\frac{\\sin(\\omega)}{\\omega} - 3\\cos(2\\omega) + 2\\cos(\\omega) \\right.\\\\&\\left.+ \\frac{\\sin(2\\omega)}{\\omega} - \\frac{\\sin(\\omega)}{\\omega} \\right)\\\\\n&= \\frac{1}{\\omega}\\sqrt{\\frac{2}{\\pi}}\\left(\\cos(\\omega) - 3\\cos(2\\omega) + \\frac{\\sin(2\\omega)}{\\omega}\\right) \n\\end{aligned}"

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!