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Question: Area of the region bounded by the curves y = 2<sup>x</sup>, y = 2x – x<sup>2</sup>, x = 0 and x = 2 ...

Area of the region bounded by the curves y = 2x, y = 2x – x2, x = 0 and x = 2 is given by

A

3log⁡2−43\frac { 3 } { \log 2 } - \frac { 4 } { 3 }

B

3log⁡2+43\frac { 3 } { \log 2 } + \frac { 4 } { 3 }

C

3 log 2 – 43\frac { 4 } { 3 }

D

3log⁡2−43\frac { 3 } { \log 2 } - \frac { 4 } { 3 }

Answer

3log⁡2−43\frac { 3 } { \log 2 } - \frac { 4 } { 3 }

Explanation

Solution

Area = ∫022x−(2x−x2)dx\int _ { 0 } ^ { 2 } 2 ^ { x } - \left( 2 x - x ^ { 2 } \right) d x = [2xlog⁡2−x2+x33]02\left[ \frac { 2 ^ { x } } { \log 2 } - x ^ { 2 } + \frac { x ^ { 3 } } { 3 } \right] _ { 0 } ^ { 2 }