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, we need to determine the points of intersection and integrate the difference between the two curves over that interval. In this case, we have two curves:
Curve 1: y = x^2
Curve 2: y = x + 2
To find the points of intersection, we set the two equations equal to each other:
x^2 = x + 2
Rewriting into standard form: x^2 - x - 2 = 0, we can factor as (x - 2)(x + 1) = 0. Solving for x, we find two points of intersection, x = -1 and x = 2.
Now we need to determine which curve is above the other over the interval between these points. To do this, we can compare the y-values of the curves at these points:
For x = -1:
Curve 1: y = (-1)^2 = 1
Curve 2: y = -1 + 2 = 1
For x = 2:
Curve 1: y = 2^2 = 4
Curve 2: y = 2 + 2 = 4
Both curves have the same y-values at the points of intersection, so we need to integrate the difference between the curves over the interval [-1, 2].
To find the area, we calculate the definite integral of (curve2 - curve1) with respect to x over the interval [-1, 2]:
Area = ∫[a,b] (curve2 - curve1) dx
= ∫[-1,2] (x + 2 - x^2) dx
Evaluating this integral will give us the area of the region between the two curves.