find the area enclosed by the curves y=absolute value of x and y=x^2+2
First use algebra to find where the enclosed area is. Then integrate the difference between the functions from one intersection to the other.
Drawing a graph will help a lot.
It seems to me, after drawing a sketch, that the two curves never intersect, and that the "enclosed area" between them is infinite. Are you sure the second curve is not x^2 -2 ? In that case they would intersect at -2 and +2
To find the area enclosed by the curves y = |x| and y = x^2 + 2, we need to determine the points of intersection between the two curves and then calculate the areas bounded by them.
Step 1: Find the points of intersection:
Set the two equations equal to each other and solve for x:
|x| = x^2 + 2
Break it down into two separate cases, considering the positive and negative values of x separately.
Case 1: x is positive:
x = x^2 + 2
Rearrange the equation: x^2 - x + 2 = 0
Solve the quadratic equation using the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(1)(2))) / (2(1))
Simplify: x = (1 ± √(-7)) / 2
Since the square root of a negative number is not a real number, x cannot be positive in this case. Therefore, there are no points of intersection in the positive x region.
Case 2: x is negative:
-x = x^2 + 2
Rearrange the equation: x^2 + x + 2 = 0
Solve the quadratic equation using the quadratic formula:
x = (-1 ± √(1^2 - 4(1)(2))) / (2(1))
Simplify: x = (-1 ± √(-7)) / 2
Similarly, since the square root of a negative number is not a real number, x cannot be negative in this case. Therefore, there are no points of intersection in the negative x region.
As a result, there are no points of intersection between the two curves.
Step 2: Determine the bounded area:
Since there are no points of intersection between the curves y = |x| and y = x^2 + 2, there is no enclosed area. The two curves do not intersect, so there is no region between them to calculate the area.
Therefore, the area enclosed by the curves is zero.