A possible sketch goes like this.

The area of the region we want is basically the area of the red region in the picture (which is enclosed by the curve and the horizontal line).

The red region is not touching the x-axis directly, so we cannot just take the area under the curve, since the area under the curve from x = 1 to x = 4 is the entire shaded region of both the red and the blue regions (under the curve ONTO the x-axis).

The area which we are looking
= Area under curve - Blue rectangle
= INT (-x² + 5x + 4) dx [upper is x = 4, lower is x = 1] - [3 x 8]
= [-x³/3 + 5x²/2 + 4x] [upper is x = 4, lower is x = 1] - 24
= [-4³/3 + 5 (4)²/2 + 4 (4)] - [-1³/3 + 5 (1)²/2 + 4 (1)] - 24
= [-64/3 + 40 + 16] - [-1/3 + 5/2 + 4] - 24
= 104/3 - 37/6 - 24
= 9/2 units²