Consider the region above the graph of y=x^2 and below the graph of y=x+2. What is the area of this region?
To find the area of the region between two curves, such as the region above the graph of y = x^2 and below the graph of y = x + 2, we need to determine the points where the two curves intersect. Then, we integrate the difference between the two curves over the interval between these points.
To find the intersection points, we set the two equations equal to each other:
x^2 = x + 2
Rearranging the equation:
x^2 - x - 2 = 0
Now, we solve for x by factoring or using the quadratic formula. In this case, we can factor it:
(x - 2)(x + 1) = 0
Setting each factor equal to zero:
x - 2 = 0 --> x = 2
x + 1 = 0 --> x = -1
So, the two curves intersect at x = 2 and x = -1.
To find the area of the region, we need to integrate the difference between the two curves over the interval [-1, 2]:
Area = ∫[a, b] (x + 2) - x^2 dx,
where a = -1 and b = 2.
Integrating the difference between the two equations:
Area = ∫[-1, 2] (x + 2) - x^2 dx
= ∫[-1, 2] x + 2 - x^2 dx
Evaluating the integral:
Area = [x^2/2 + 2x - (x^3/3)] from -1 to 2
= [(2^2/2 + 2(2) - (2^3/3)) - ((-1)^2/2 + 2(-1) - ((-1)^3/3))]
= [2 + 4 - (8/3)] - [1/2 - 2 + (1/3)]
Simplifying further:
Area = 28/3 - 1/2 + 1/3
Therefore, the area of the region between the two curves is:
Area = 16/3 - 1/2 = 15/2 = 7.5 square units.