Question

Determine the area of the region bounded by the given set of curves y=x√x^2 +1, y=e^-1/2, x=-3 and the y-axis.

Answer 1

The region bounded by equation $y=x√(x^2+1)$ , $y=e^{-1/2}$ , $x=-3$ and y-axis

$\therefore area= ∫^{0} _{-3} [(e) ^{-1/2} -x√(x^{2}+1)) ]dx$

$=∫_{-3}^{0}e^{-1/2}dx- ∫_{-3}^{0}x√(x^2+1 ) dx$

$=[ e^{-1/2} x]|_{-3}^{0}-1/2 ∫_{-10}^{1} u^{1/2} du$

letting $u=x^2+1$ in the second integral and substituting up to limits 

$=[0-e^{-1/2} (-3)]- \frac{1}{2} [\frac{2}{3}u^{3/2} )]|^{1} _{10}$

$=3e^{-1/2}- \frac{1}{3} [1^{3/2 } -10^{3/2} ]$

$=3e^{-1/2}+ \frac{10}{3} √10-\frac{1}{3}$

 = 12.03 square units of length