a) $\int {{{\left( {x + 1} \right)}^{\frac{1}{2}}}}  - \left( {x + 1} \right)$ or

$\int {\left( {{y^2} - 1} \right)}  - \left( {y - 1} \right)$

$\frac{2}{3}{\left( {x + 1} \right)^{\frac{3}{2}}} - \frac{1}{2}{x^2} - x$

or $\frac{1}{3}{y^2} - \frac{1}{2}{y^2}$

$\frac{2}{3} - \left( {0 - \frac{1}{2} + 1} \right)$ or $\frac{1}{3} - \frac{1}{2}$

$\frac{1}{6}$

b) ${V_1} = \left( \pi  \right)\int {{{\left( {{y^2} - 1} \right)}^2}}  = \left( \pi  \right)\int {{y^4} - 2{y^2} + 1} $

$\left( \pi  \right)\left[ {\frac{{{y^5}}}{5} - \frac{{2{y^2}}}{3} + y} \right]$

$\left( \pi  \right)\left[ {\frac{1}{5} - \frac{2}{3} + 1} \right]$

${V_1} = \frac{8}{{15\left( \pi  \right)}}$ or $0.533\left( \pi  \right)$ (AWRT)

or $\left( \pi  \right)\left[ {\frac{{{y^3}}}{3} - {y^2} + y} \right]$

${V_2} = \frac{1}{3}\pi $

Volume $ = \frac{8}{{15}}\pi \frac{1}{{ - 3}}\pi  = \frac{1}{5}\pi $ (or $0.628$)

OR $\left( {{y^4} - 2{y^2} + 1} \right) - \left( {{y^2} - 2y + 1} \right)$

$\left( \pi  \right)\int {{y^4} - 3{y^2} + 2y} $

$\left( \pi  \right)\left[ {{y^ \uparrow }5/5 - {y^ \uparrow }3 + {y^ \uparrow }2} \right]$

$\left( \pi  \right)\left[ {\frac{1}{5} - 1 + 1} \right]$

$\frac{1}{5}\pi $

$\int\limits_{ - 1}^0 {x + 1 - } \int\limits_{ - 1}^0 {{{\left( {x + 1} \right)}^2}} $

$\left[ {\frac{{{x^2}}}{2} + x} \right] - \left[ {\frac{{x + {1^3}}}{3}} \right]$

$SC = \left[ {\left( 0 \right) - \left( {\frac{1}{2} - 1} \right)} \right] - \left[ {\frac{1}{3} - 0} \right]$

$\frac{1}{2} - \frac{1}{3} = \frac{1}{6}\pi \,\,\,\left( {0.524} \right)$