first, we find where the curve touches the x-axis, or simply get the roots,, to get the roots, equate y = 0:
0 = 4x - x^2
0 = x(4-x)
x = 0 and x = 4
this will be our boundary conditions for integration,, we'll use integration since area under the curve:
integral (4x - x^2) = 2x^2 - (1/3)X^3
at x = 0, the integral becomes
2*0^2 - (1/3)*0^3 = 0
at x = 4 the integral becomes
2*(4^2) - (1/3)*4^3
32 - 64/3
96/3 - 64/3
32/3 = 10 2/3
thus integral from 0 to 4 = 10 2/3 - 0 = 10 2/3
hope this helps~ :)
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