A) \[37\pi /3\]
B) \[7\pi \sqrt{2}\]
C) \[37\pi \]
D) \[y={{x}^{2}}\]
Correct Answer: B
Solution :
Curved surface \[=\int_{a}^{b}{2\pi \,y\sqrt{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right]}dx}\] Given that \[a=2,\]\[b=3\] and \[y=x+1\]. On differentiating with respect to \[x\], \[\frac{dy}{dx}=1+0\,\,\,\text{or}\,\,\frac{dy}{dx}=1\] Therefore, curved surface \[=\int_{2}^{3}{2\pi (x+1)\sqrt{[1+{{(1)}^{2}}]\,}dx}\]\[=\int_{2}^{3}{2\pi (x+1)\sqrt{2\,}}dx\] \[=2\sqrt{2\,}\pi \int_{2}^{3}{(x+1)\,}dx\]\[=2\sqrt{2\,}\pi \left[ \frac{{{(x+1)}^{2}}}{2} \right]_{2}^{3}\] \[=\frac{2\sqrt{2}}{2}\pi [{{(3+1)}^{2}}-{{(2+1)}^{2}}]=\sqrt{2}\pi (16-9)=7\sqrt{2}\pi =7\pi \sqrt{2}\].
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