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find the area bounded by the curve y=2x-x^2 and the straight line y=-x

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Please find below the solution to the asked query:


y=2x-x2 ;iy=-xPutting y=-x in i, we get,-x=2x-x2⇒x2-3x=0⇒xx-3=0⇒x=0 and x=3Nowy1=2x-x2 y2=-xAreaA=∫03y1-y2.dx=∫032x-x2--x.dx=∫032x-x2+x.dx=∫033x-x2.dx=3x22-x3303=3322-333=272-273=2712-13=276⇒Area=276 unit2


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y =2x - x^2 & y =-x

y= x (2-x)

Taking y = 0 for the curve
we get x = 0 or x = 2
(0,0) & (2,0) are the two points on the curve. Now as both the points are on X - Axis let us take a point x = 1 then we will get y = 1
So we have 3 points
(0,0) , (2,0) & (1,1)

Joining this points you will get a parabola of inverted you shape.

y = -x is a line passing through origin in second & fourth quadrant
you can take (0,0) , (1,-1), (-1,1) points to plot the line

now find the common region .

solve the equations of curve & line simultaneously & Integrate the common region !
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