find the area of the region bounded by the graphs of y=x^2 and y=cos(x)
The curves intersect at x = ±0.824
Since the area is symmetric we can just work with one half of it, and double that value:
a = 2∫[0,0.824] cosx - x^2 dx
So the answer would be 1.0947?
To find the area of the region bounded by the graphs of y=x^2 and y=cos(x), we need to determine the points of intersection between the two graphs. The area of the region between two curves can be found by taking the definite integral of the difference of the two functions over the interval where they intersect.
First, let's find the points of intersection:
Setting the two functions equal to each other, we have:
x^2 = cos(x)
To solve this equation, we can use either numerical methods or graphical analysis.
Let's use graphical analysis to find the approximate points of intersection:
1. Graph the two functions y = x^2 and y = cos(x) on the same coordinate system.
2. Look for the points where the two graphs intersect. These points will give us the x-coordinates of the points of intersection.
Based on the graph, we can see that there are four points of intersection: (0, 0), (1, 0.5403), (-1, 0.5403), and (1.763, 1.0002).
Next, we need to determine the interval over which we will evaluate the definite integral. From the graph, it is evident that the functions intersect in the interval [-1, 1].
Now, we can set up the definite integral to find the area:
Area = ∫[a, b] (x^2 - cos(x)) dx
where a and b are the x-coordinates of the points of intersection, which in this case are a = -1 and b = 1.
Calculating the definite integral will give you the area of the region bounded by the two graphs.