# How do you find the area cut off by the x axis, above the x-axis, and #y=(3+x)(4-x)#?

Start by expanding.

We need to evaluate the following integral:

Use the second fundamental theorem of calculus to evaluate.

Hopefully this helps!

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To find the area cut off by the x-axis, above the x-axis, and the curve ( y = (3 + x)(4 - x) ), you need to first identify the x-values where the curve intersects the x-axis. These are the roots of the equation ( (3 + x)(4 - x) = 0 ).

To find the roots, set ( (3 + x)(4 - x) = 0 ) and solve for ( x ).

( (3 + x)(4 - x) = 0 )

This equation yields two solutions for ( x ).

Once you find the x-values where the curve intersects the x-axis, you can integrate the absolute value of the function within the given range to find the area cut off by the x-axis above the x-axis. This is because the absolute value of the function will ensure that the area is positive. Use the formula for the area under a curve:

[ \text{Area} = \int_{a}^{b} |f(x)| , dx ]

where ( a ) and ( b ) are the x-values where the curve intersects the x-axis. Integrate ( |(3 + x)(4 - x)| ) with respect to ( x ) from the smaller root to the larger root to find the area.

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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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