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How do I find the area between the curves #y=x^2-4x+3# and #y=3+4x-x^2#?

Answer 1

William Abdalla

It helps to first make a scetch of the two curves. Preferably on the same paper (can't get this done here)

Then find the intersection points where both are equal: #y=x^2-4x+3=-x^2+4x+3#

You will find #x=0# and #x=4#

Now we need the difference-function (the space between the functions): #y=(x^2-4x+3)-(-x^2+4x+3)=2x^2-8x#

Then we integrate this function from #0# to #4#

Area = #int_0^4 (2x^2-8x).dx=|_0^4 2/3x^3-4x^2=#

Area = #|(2/3*4^3-4*4^2)-0|=21 1/3#

(we take the absolute, positive value)

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Answer 2

Axel Alvarez

To find the area between the curves (y = x^2 - 4x + 3) and (y = 3 + 4x - x^2), you first need to determine the points of intersection of these two curves by setting them equal to each other and solving for (x). Then, integrate the absolute difference of the two functions over the interval of intersection. This will give you the area between the curves.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.