Calculate the area between the curve y=3x-x^2 and the x axis between x=10 and x= 6?

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Answer 1 · 2018-01-13T23:12:25

#A=493/3#

#y=3x-x^2#

The graph of this function looks like this.

graph{3x-x^2 [-3.46, 11, -100, 10]}

All of the area between #x=6# and #x=10# is below the axis so we do need to correct for this. The area is given by:

#int_6^10 3x-x^2dx=[3/2x^2-1/3x^3]_6^10#

Now evaluate the limits:

#{3/2(10)^2-1/3(10)^3}-{3/2(6)^2-1/3(6)^3}#

#=3/2(100)-1/3(1000)-3/2 36+1/3 216#

#=-496/3#

It is negative because the area is under the curve, we can simply to take the positive to get:

#A=493/3#