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Question: The value of \(\int _ { 0 } ^ { 2 } \left| x ^ { 3 } - 3 x ^ { 2 } + 2 x \right|\) dx is...

The value of ∫02∣x3−3x2+2x∣\int _ { 0 } ^ { 2 } \left| x ^ { 3 } - 3 x ^ { 2 } + 2 x \right| dx is

A

12\frac { 1 } { 2 }

B

14\frac { 1 } { 4 }

C

116\frac { 1 } { 16 }

D

None of these

Answer

12\frac { 1 } { 2 }

Explanation

Solution

x3 – 3x2 + 2x = 0

Ž x (x2 – 3x + 2) = 0

Ž x (x – 2) (x – 1) = 0

x = 0, 1, 2.

∫02∣x3−3x2+2x∣dx\int _ { 0 } ^ { 2 } \left| x ^ { 3 } - 3 x ^ { 2 } + 2 x \right| d x

= ∫01(x3−3x2+2x)dx−∫12(x3−3x2+2x)dx\int _ { 0 } ^ { 1 } \left( x ^ { 3 } - 3 x ^ { 2 } + 2 x \right) d x - \int _ { 1 } ^ { 2 } \left( x ^ { 3 } - 3 x ^ { 2 } + 2 x \right) d x

= [x44−x3+x2]01\left[ \frac { x ^ { 4 } } { 4 } - x ^ { 3 } + x ^ { 2 } \right] _ { 0 } ^ { 1 }–