What is the net area between #f(x) = sqrtx-x # and the x-axis over #x in [2, 4 ]#?

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Answer 1 · 2017-05-30T16:37:56

#"Area" ~~ -2.5523#

Here is a graph of #f(x)#:

graph{sqrtx - x [-10, 10, -5, 5]}

Please observe that #f(x)# is negative over the interval, therefore, the area will be negative.

#"Area" = int_2^4 x^(1/2)-x dx#

#"Area" = (2/3x^(3/2)-1/2x^2]_2^4#

#"Area" = (2/3(4)^(3/2)-1/2(4)^2) - (2/3(2)^(3/2)-1/2(2)^2)#

#"Area" ~~ -2.5523#