find the area of the region bounded by the function y=-x^2+5x-4 and the x-axis

Answer 1

y = -(x-1)(x-4)

so you want to integrate between the roots, in the interval [1,4]

a = ∫[1,4] -x^2+5x-4 dx
= -1/3 x^3 + 5/2 x^2 - 4x [1,4]

Now just evaluate at the ends and subtract.