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Question

Question: The area bounded by the curve \(y = 2x - x^{2}\) and the straight line \(y = - x\) is given by...

The area bounded by the curve y=2xx2y = 2x - x^{2} and the straight line y=xy = - x is given by

A

9/2

B

43/6

C

35/6

D

16/3

Answer

9/2

Explanation

Solution

Required Area = 03(y1y2)dx\int_{0}^{3}{\left( y_{1} - y_{2} \right)dx}

=03(2xx2)(x)dx=03(3xx2)dx\int_{0}^{3}{\left( 2x - x^{2} \right) - ( - x)dx = \int_{0}^{3}{\left( 3x - x^{2} \right)dx}}

= [3x22x33]03=2729=92\left\lbrack \frac{3x^{2}}{2} - \frac{x^{3}}{3} \right\rbrack_{0}^{3} = \frac{27}{2} - 9 = \frac{9}{2}