# What is the net area between #f(x)=7x-8# in #x in[1,2] # and the x-axis?

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To find the net area between the function ( f(x) = 7x - 8 ) and the x-axis over the interval ([1,2]), you need to compute the definite integral of ( f(x) ) from ( x = 1 ) to ( x = 2 ) and take the absolute value of the result. This is because the area between a function and the x-axis can be negative if the function dips below the x-axis. So, to get the net area, we ensure it's positive.

The definite integral of ( f(x) = 7x - 8 ) from ( x = 1 ) to ( x = 2 ) is:

[ \int_{1}^{2} (7x - 8) , dx ]

This integral evaluates to:

[ \left[ \frac{7}{2}x^2 - 8x \right]_{1}^{2} ]

[ = \left( \frac{7}{2}(2)^2 - 8(2) \right) - \left( \frac{7}{2}(1)^2 - 8(1) \right) ]

[ = \left( \frac{7}{2}(4) - 16 \right) - \left( \frac{7}{2}(1) - 8 \right) ]

[ = \left( 14 - 16 \right) - \left( \frac{7}{2} - 8 \right) ]

[ = (-2) - \left( \frac{7}{2} - 8 \right) ]

[ = -2 - \frac{7}{2} + 8 ]

[ = -2 - \frac{7}{2} + \frac{16}{2} ]

[ = -\frac{4}{2} - \frac{7}{2} + \frac{16}{2} ]

[ = -\frac{11}{2} + \frac{16}{2} ]

[ = -\frac{11}{2} + 8 ]

[ = -\frac{11}{2} + \frac{16}{2} ]

[ = \frac{5}{2} ]

Taking the absolute value, the net area between the function ( f(x) = 7x - 8 ) and the x-axis over the interval ([1,2]) is ( \frac{5}{2} ).

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