question_answer

The value of \[\int_{0}^{8}{|x-5|}\,\,dx\]is:

A) \[9\]                                     

B) \[12\]

C) \[17\]                                   

D) \[18\]

Correct Answer: C

Solution :

Let\[I=\int_{0}^{8}{|x-5|}\,\,dx\]         \[=\int_{0}^{5}{(5-x)\,}dx+\int_{5}^{8}{(x-5)}\,dx\]                 \[=\left[ 5x-\frac{{{x}^{2}}}{2} \right]_{0}^{5}+\left[ \frac{{{x}^{2}}}{2}-5x \right]_{5}^{8}\]                 \[=25-\frac{25}{2}+\left[ \frac{64}{2}-40-\left( \frac{25}{2}-25 \right) \right]\]                 \[=\frac{25}{2}-8+\frac{25}{2}=17\] Alternative Solution: It is clear from [he figure that required area = area of\[\Delta OAD+\]area of\[\Delta ABC\]                 \[=\frac{1}{2}\times 5\times 5+\frac{1}{2}\times 3\times 3\]                 \[=\frac{34}{2}=17\]