Question

What is the net area between `f(x) = x-xsqrt(4x-3)` and the x-axis over `x in [1, 4 ]`?

Answer 1

`A=int_1^4[0-(x-xsqrt(4x-3))]*dx=int_1^4-x+xsqrt(4x-3)*dx=13.44 (unite)^2`

Explanation:

show below:

the sketch of the function `f(x) = x-xsqrt(4x-3)`
graph{x-xsqrt(4x-3) [-10, 10, -5, 5]}

the area wanted is between x=1 and x=4 as shown in the graph below

now , set up the integral

`A=int_1^4[0-(x-xsqrt(4x-3))]*dx=int_1^4-x+xsqrt(4x-3)*dx`

`=[(4*x-3)^(5/2)/40+(4*x-3)^(3/2)/8-x^2/2]_1^4=[(9*13^(3/2)-153)/20]=13.44(unite)^2`