Question

Could someone show me a step by step process on how to do this problem? Calculus 2

Answer 1

The arc length is given by the definite integral

`\displaystyle \int_1^3 √(1 + \left(y'\right)^2) \, dx = \int_1^3 √(1+9x) \, dx`

since by the power rule for differentiation,

`y = 2x^(3/2) \implies y' = \frac32 \cdot 2x^(3/2-1) = 3x^(1/2) \implies \left(y'\right)^2 = 9x`

To compute the integral, substitute

`u = 1+9x \implies du = 9\,dx`

so that by the power rule for integration and the fundamental theorem of calculus,

`\displaystyle \int_(x=1)^(x=3) √(1+9x) \, dx = \frac19 \int_(u=10)^(u=28) u^(1/2) \, du = \frac19*\frac23 u^(1/2+1) \bigg|_(10)^(28) = \boxed{\frac2{27}\left(28^(3/2) - 10^(3/2)\right)}`