Compute the area bounded by the curve y=x4−2x3+x2+3, the axis of abscissa and two ordinates corresponding to the points of minima of the function y(x)
A
9130 square units
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B
9330 square units
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C
130 square units
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D
None of these
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Solution
The correct option is A9130 square units The given curve is y=x4−2x3+x2+3, since f is differentiable function
⇒dydx=4x3−6x2+2x=2x(2x2−3x+1)
⇒ Thus x=0,12,1 are critical points
⇒d2ydx2=12x2−12x+2
⇒ Now taking the sign of d2ydx2 at the critical point we get
(d2ydx2)x=0=2,(d2ydx2)x=12=−1,(d2ydx2)x=1=2
Curve has minima at x=0 and x=1:f(−∞)=∞,f(0)=3,f(1)=3,f(∞)=∞
∴f(x)>0∀xeR
∴ Required area =∫10ydx=∫10(x4−2x3+x2+3)dx=[x55−x22+x33+3x]10=9130 square units