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\documentclass[10pt]{article}
\usepackage[top=.5in, bottom=.5in,left=0.5in,right=0.5in]{geometry}
\usepackage{amsmath}
\usepackage{pstricks-add}
\begin{document}
The region is rotated around the x-axis, find the volume bounded by: $\displaystyle \quad y=(\,x+1\,)^{2}, \quad y{\,=\,}0, \quad x=1, \quad x=2 $\\\\\\
\begin{center}
\newrgbcolor{zzttqq}{0.6 0.2 0}
\psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-0.5,-0.5)(2.5,9.3)
\psaxes[labelFontSize=\scriptstyle,xAxis=true,yAxis=true,Dx=1,Dy=1,ticksize=-2pt 0,subticks=2]{->}(0,0)(-0.5,-0.5)(2.5,9.3)[x,140] [y,-40]
\pscustom[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1]{\psplot{1}{2}{(x+1)^2}\lineto(2,0)\lineto(1,0)\closepath}
\psplot[plotpoints=200]{-0.5}{2.5}{(x+1)^2}
\psline(1,-0.5)(1,4)
\rput[tl](0.1,8.1){$y=(x+1)^{2}$}
\psline(2,-0.5)(2,10)
\end{pspicture*}
\newline
\end{center}
\begin{align*}
\text{Area} &= \int_a^{b} \Bigg[\, \pi\,(\text{\,radius\,})^{2}\; \Bigg] \,dx \\\\
~ &= \int_0^{1} \Bigg[\, \pi\,(\,x^{2}\,)^{2}\; \Bigg] \,dx \\\\
~ &= \int_0^{1} \Bigg[\, \pi\,x^{4}\; \Bigg] \,dx \\\\
~ &= \Bigg[\, \pi\,\frac{\,x^{5}}{5} \; \Bigg]_0^{1} \\\\
~ &= \Bigg[\, \pi\,\frac{\,1^{5}}{5} \; - \pi\,\frac{0^{5}}{4} \, \Bigg] \\\\
~ &= \,\frac{\pi}{5} \, - \, 0 \\\\
~ &= \frac{\pi}{5}
\end{align*}
\end{document}
