Mathematicity
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- Jun 4, 2022
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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.
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.