A) \[\frac{20\pi a{{R}^{4}}}{331}\]
B) \[\frac{15\pi a{{R}^{4}}}{512}\]
C) \[\frac{\pi a{{R}^{4}}}{332}\]
D) \[\frac{3\pi a{{R}^{4}}}{864}\]
Correct Answer: B
Solution :
Given \[J=a{{r}^{2}}\] As \[I=\int{J\,dA}\] Here, \[dA=2\pi rdr\] \[[\because A=\pi {{r}^{2}}]\] So \[I=\int_{R/4}^{R/2}{a{{r}^{2}}(2\pi r)dr}\] \[I=2\pi a\int\limits_{R/4}^{R/2}{{{r}^{3}}dr}=2\pi a\left[ \frac{{{r}^{4}}}{4} \right]_{R/4}^{R/2}\] \[=\frac{\pi a}{2}\left[ {{\left( \frac{R}{2} \right)}^{4}}-{{\left( \frac{R}{4} \right)}^{4}} \right]\] \[=\frac{\pi a{{R}^{4}}}{2}\left[ \frac{1}{16}-\frac{1}{256} \right]\] \[=\frac{\pi a{{R}^{4}}}{2}\left[ \frac{256-16}{4096} \right]\] \[=\frac{15\pi a{{R}^{2}}}{512}\]
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