Question

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

Answer 1

`[(1/2-5/6sqrt5+1/12sqrt5)-(8-34/3sqrt17+17/60sqrt17)]=36.38` areal units, nearly. Please do not attempt to edit this answer. I would review my answer, for bugs, if any, myself, after some time.

Explanation:

The integrated area is in y-negative `Q_4` and is negative.

The end points for the curved boundary are

A at `(1, -(sqrt5-1))=(1. -1.236)` and

B at `(4,=4(sqrt17-1))=(4, =12.402)`.

The numerical area ( suppressing negative sign )

`=- int y dx`, with `y = x(1-sqrt(4x+1))` and x, from 1 to 4

`=-int(x-xsqrt(4x+1)) dx`, for the limits 1 and 4

`=-[x^2/2-(2/3)(1/4)int x d(4x+1)^(3/2)]`, for x = 1 to 4

#=-[x^2/2-1/6x(4x+1)^(3/2)+1/6int(4x+1)^(3/2) dx], for x = 1 to 4

`=-[x^2/2-1/6x(4x+1)^(3/2)+1/6(2/5)(1/4)(4x+1)^(5/2) ],`

between x = 1 and 4

`=-[(8-34/3sqrt17+17/60sqrt17)-(1/2-5/6sqrt5+1/12sqrt5)]`

graph{x(1-sqrt(4x+1)) [0, 50, -25, 0]}