Science:Math Exam Resources/Courses/MATH103/April 2016/Question 04 (a)/Solution 1

By the formula for the mass $m$ of an object given its linear density function $\rho ,$ it is enough to compute

$m=\int _{0}^{3}\rho (x)dx=\int _{0}^{3}{\frac {1}{\sqrt {x+1}}}\,dx.$

Using the substitution $u=x+1,du=dx,$ the integral becomes

${\begin{aligned}m&=\int _{0+1}^{3+1}{\frac {1}{\sqrt {u}}}\,du\\&=\int _{1}^{4}u^{\frac {-1}{2}}du\\&=\left.2u^{\frac {1}{2}}\right|_{1}^{4}\\&=2({\sqrt {4}}-{\sqrt {1}})\\&=\color {blue}2.\end{aligned}}$