Question

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 .

Answer 1

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