Find the area y^x^2 , y=4x-x^2
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Find the area y=x^2 and y=4x-x^2
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The area of the region bounded by the curve y=x^2 and y=4x-x^2
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yes, you right. It is y=x^2 , y=4x-x^2
To find the area of the region defined by the equations y = x^2 and y = 4x - x^2, we need to determine the limits of integration for the x variable. The region is bound by the points where these two curves intersect.
First, set the two equations equal to each other:
x^2 = 4x - x^2
Simplifying, we get:
x^2 = 2x
Rearranging the equation:
x^2 - 2x = 0
Factoring out an x:
x(x - 2) = 0
This equation is satisfied when x = 0 or x = 2.
Now that we know the limits of integration for the x variable, we can set up the integral to find the area. The area A is given by:
A = ∫[0,2] (y^x^2) dx
Substituting y = 4x - x^2, we get:
A = ∫[0,2] ((4x - x^2)^x^2) dx
We can simplify the integral by expanding the expression:
A = ∫[0,2] (16x^2 - 8x^3 + x^4) dx
Evaluating this integral will give us the area of the region defined by the given equations.