Find the area bounded by {y=x2−4
y=4−x2
• sketch the region described
• determine any intersection point(s) for the curves (show work!!)
• write out the integral(s) that will calculate the area
• determine the area (may use a calculator)
the parabolas intersect at x = -2,2
area is thus
∫[-2,2] (4-x^2) - (x^2-4) dx = 64/3
To find the area bounded by the curves y = x^2 - 4 and y = 4 - x^2, we can follow these steps:
1. Sketch the Region:
Start by sketching the two curves on a coordinate plane. The first curve, y = x^2 - 4, is an upward-opening parabola that is shifted 4 units downward. The second curve, y = 4 - x^2, is a downward-opening parabola that is shifted 4 units upward. Make sure to include the x and y axes to indicate the intersections.
2. Determine the Intersection Point(s):
To find the intersection point(s) of the curves, set the two equations equal to each other and solve for x:
x^2 - 4 = 4 - x^2
2x^2 = 8
x^2 = 4
x = ±2
Substituting these x-values back into either of the equations, we find the corresponding y-values:
For x = 2: y = 2^2 - 4 = 0
For x = -2: y = (-2)^2 - 4 = 0
So, the curves intersect at the points (2, 0) and (-2, 0).
3. Write Out the Integral(s):
To calculate the area between the curves, we need to set up an integral using the limits of integration. Since the curves intersect at x = -2 and x = 2, the integral can be written as follows:
Area = ∫(-2 to 2) [upper curve - lower curve] dx
The upper curve is 4 - x^2, and the lower curve is x^2 - 4. Therefore, the integral becomes:
Area = ∫(-2 to 2) (4 - x^2) - (x^2 - 4) dx
4. Determine the Area:
You can now solve the integral either by hand or using a calculator to determine the area between the curves. After evaluating the integral, you will obtain the area bounded by the curves y = x^2 - 4 and y = 4 - x^2.