Find the area of the surface that is generated by revolving the portion of the curve y = x
2
between x = 2 and x = 3 about the x-axis.
"=\\displaystyle\\int_{2}^{3}2\\pi x^2\\sqrt{1+(2x)^2}dx"
"\\int x^2\\sqrt{1+(2x)^2}dx="
"x=\\dfrac{1}{2}\\tan \\theta, -\\dfrac{\\pi}{2}<\\theta <\\dfrac{\\pi}{2}"
"dx=\\dfrac{1}{2\\cos^2\\theta}d\\theta"
"\\sqrt{1+(2x)^2}=\\dfrac{1}{\\cos\\theta}"
"\\int x^2\\sqrt{1+(2x)^2}dx=2\\int (x^2)\\sqrt{1+(2x)^2}dx"
"=\\int \\dfrac{1}{4}\\tan^2 \\theta(\\dfrac{1}{\\cos\\theta})(\\dfrac{1}{2\\cos^2\\theta})d\\theta"
"=\\dfrac{1}{8}\\int (\\dfrac{1}{\\cos^3\\theta}-\\dfrac{1}{\\cos^5\\theta})d\\theta"
"=\\dfrac{1}{8}(\\ln(|\\tan\\theta+\\sec \\theta|)+\\tan\\theta\\sec\\theta(1-2\\sec^2\\theta))+C"
"=\\dfrac{\\sqrt{1+4x^2}(16x^3+2x)-\\ln(|\\sqrt{1+4x^2}+2x|)}{64}+C"
"S=\\dfrac{\\pi}{32}\\bigg(438\\sqrt{37}-\\ln(\\sqrt{37}+6)"
"-132\\sqrt{17}+\\ln(\\sqrt{17}+4)\\bigg)\\ square\\ units"
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