How do I find the area bounded by the curve y=-x^2+4 and the line y=-x+2?
First sketch the line and the curve on the same axes (I would show this using the whiteboard).Then we want to find the points of intersection so set the two equations equal to each other and rearrange to get the equation x^2-x-2=0. If we solve this for x and substitute the values of x back into one of the equations we find that the points of intersection are at (-1,3) and (2,0). Then to find the area under the curve we can integrate -x2+4 between x=-1 and x=2 to get 9 units squared. But this is not the area we want as it includes the area under the line. So we need to subtract the area under the line between x=-1 and x=2. From the sketch this is the same as subtracting the area of the triangle with vertices at (-1,3),(-1,0) and (2,0) which equals 9/2. So the final solution is 9-9/2 = 9/2 units squared.
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