Question

find the arc length of the curve y=x^3/2-1 from x=0 to x=1​

Answer 1

Answer:

$s = \int_{0} ^{1} ( \sqrt{1 + ( \frac{dy}{dx} } ) {}^{2} dx \\ \\ \: \: \: = \int_{0} ^{1} \sqrt{1 + \frac{3 {x}^{2} }{2} } dx \\ \\ = \frac{1}{ \sqrt{2} } \int_{0} ^{1} \sqrt{( \sqrt{2} ) {}^{2} + {( \sqrt{3x)} }^{2} } dx \\ \\ = \frac{1}{ \sqrt{2} } ( \frac{x \sqrt{2 + 3 {x}^{2} } }{2} + \frac{2}{2} ln(x + \sqrt{2 + 3 {x}^{2} } ) \\ \\ = \frac{1}{ \sqrt{2} } ( \frac{ \sqrt{5} }{2} + ln(1 + \sqrt{5} ) - ln( \sqrt{2} ) )$