Answer to Question #249725 in Calculus for SHAIKAT

Question #249725

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.


1
Expert's answer
2021-10-12T02:47:18-0400
"S=\\displaystyle\\int_{2}^{3}2\\pi y\\sqrt{1+(y')^2}dx"

"=\\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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