Area under curve question

[Mathematicity]

Hi, this is the question:

Figure 3 shows part of the curve with the equation y= (x^3)-5(x^2) +2x + 8 and the line AB. A has coordinates (0,0). B has coordinates (1,6) and lies on the curve.
(-1,0) and C are two points of intersection between the curve and the x-axis.

a) Find the co-ordinates of point C. I have understood this and got the correct answer. C=(2,0).

b) Find the exact area of the shaded region. The shaded region is what seems to be a triangle with coordinates: A=(0,0), B=(1,6) and C=(2,0).
My thinking is that we know the length AC= 2 and the perpendicular height= 6, so using (1/2)base*height= 6. Therefore area of shaded region = 6 units^2.

However, the answer sheet has done the following:
Shaded area=
((1/2)*1*6) - (integrating: (x^3)-5(x^2) +2x + 8)dx, upper limit=2, lower limit=1).
=73/12.
So it looks like the shaded area is: area of triangle subtract integral of the curve with limits of 2 and 1.

Does the question have a mistake?
Why is the area of the shaded region not just simply the area of the triangle?

Thank you for your help in advance.

 

[Dr.Peterson]

Why is the area of the shaded region not just simply the area of the triangle?

Thank you for your help in advance.

It would have made things easier if you had included an image; but here is mine:

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Do you not see that from B to C, the region is bounded by the given curve, not by a straight line?

Here I have added the line you used:

​

They are not the same, so your answer, while close, is not exact.

 

[Mathematicity]

They are not the same, so your answer, while close, is not exact.

Thanks a lot for your help.
That explains why the actual answer, 73/12 which is approx. 6.08333 is close to the area of the triangle of 6.