# How do you find the area between #f(x)=-x^2+4x+1, g(x)=x+1#?

Please see the explanation.

Find the boundaries of the area by setting

This is confirmed by the graph of the two functions:

Let the lower limit of integration

Let the upper limit of integration

The area between the two fuctions is:

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To find the area between the curves ( f(x) = -x^2 + 4x + 1 ) and ( g(x) = x + 1 ), first, determine their intersection points by setting ( f(x) = g(x) ) and solving for ( x ). Then, integrate the absolute difference between the two functions over the interval between these intersection points. The area can be calculated as ( \int_{a}^{b} |f(x) - g(x)| , dx ), where ( a ) and ( b ) are the x-coordinates of the intersection points.

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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.

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.

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.

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