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

Answer 1

#343/6# square units.

Start by expanding.

#y = -x^2 + 4x - 3x + 12#
#y = -x^2 + x + 12#
This is a parabola that opens downward. So, the area we need to find is between #a# and #b#, where #a# and #b# are the zeroes of your function.
We started in factored form, so we can immediately say that our zeroes are #x = -3# and #x= 4#.

We need to evaluate the following integral:

#int_-3^4 -x^2 + x + 12 dx#
#=int_-3^4 -1/3x^3 + 1/2x^2 + 12x #

Use the second fundamental theorem of calculus to evaluate.

#=-1/3(4)^3 + 1/2(4)^2 + 12(4) - (-1/3(-3)^3 + 1/2(-3)^2 + 12(-3))#
#= -64/3 + 8 + 48 - 9 - 9/2 +36#
#=11 - 9/2 - 64/3#
#=83 - 9/2 - 64/3#
#=343/6# square units

Hopefully this helps!

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Answer 2

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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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.

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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