Final answer:
To find the area enclosed by the curves y = x² - 4x + 10 and y = -x² + 58, solve for the intersection points using algebraic methods, then integrate the difference of the two functions between the intersection points to find the area.
Step-by-step explanation:
To find the area enclosed by the curves y = x² - 4x + 10 and y = -x² + 58, we first need to determine the points of intersection by setting the two equations equal to each other:
x² - 4x + 10 = -x² + 58
2x² - 4x - 48 = 0
x² - 2x - 24 = 0
This is a quadratic equation of the form ax² + bx + c = 0. We can solve for x using the quadratic formula:
x = √(±2² - 4(1)(-24)) / (2(1))
= √(4 + 96) / 2
= √(100) / 2
= 10 / 2
x = 5 or x = -2
Now that we have the points of intersection, we can find the area between the curves by integrating the difference between the two functions from the lower bound of intersection (x = -2) to the upper bound (x = 5):
∫(5, -2) (-x² + 58) - (x² - 4x + 10) dx
This will give us the enclosed area. Since this is a High School level math problem involving calculus concepts, it requires familiarity with algebraic manipulations and the fundamental theorem of calculus to compute the desired area. Remember, in problems involving areas between curves, it's crucial to identify intersection points as the limits of integration.