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Manipal MET2013MathematicsDefinite Integration

The value of _a^b |x| x d x is

Options

  1. A|b|-|a|
  2. B|a|-|b|
  3. C|b|+|a|
  4. D-|b|-|a|

Correct answer

A. |b|-|a|

Step-by-step solution

Case I. If 0 a b , then |x| x =1 I= _a^b 1 d x=b-a=|b|-|a| Case II: If a b 0 , then |x|=-x I= _a^b -x x d x= _a^b(-1) d x=[-x]_a^b=-b-(-a)=|b|-|a| Case III If a 0 b . then |x|=-x when a x 0 and |x|=x when 0 x bI= _a^b |x| x d x = _a^0 |x| x d x+ ₀^b |x| x d x aligned & = _a^0 -x x d x+ ₀^b x x d x & = _a^0(-1) d x+ ₀^b(1) d x & =[-x]_a^0+[x]₀^b & =a+b=b-(-a) & =|b|-|a| aligned Hence, in all cases, I= _a^b |x| x d x=|b|-|a|

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