Answer to Question #244155 in Calculus for Genno

Question #244155
Find the length of the arc of the curve y^2b= 4/9 (1+x^2)^3 in the first quadrant from the point where x = 0 to the point where x = 3 .
1
Expert's answer
2021-09-30T00:53:32-0400

"y^2=\\frac{4}{9}(1+x^2)^3\\\\\ny=\\frac{2}{3}(1+x^2)^\\frac{3}{2}\\\\\ny'=2x(1+x^2)^\\frac{1}{2}\\\\\nL=\\int^{3}_{0}\\sqrt{1+(2x(1+x^2)^\\frac{1}{2})^2}dx=\\\\\n=\\int^{3}_{0}\\sqrt{1+4x^2(1+x^2)}dx=\\\\\n=\\int^{3}_{0}\\sqrt{1+4x^2+4x^4}dx=\\\\\n=\\int^{3}_{0}\\sqrt{(1+2x^2)^{2}}dx=\\\\\n=\\int^{3}_{0}(1+2x^2)dx=(x+\\frac{2}{3}x^3)|^3_0=3+\\frac{2}{3}\\cdot 27=21"


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