Answer :
Final answer:
The area of the region enclosed between the curves y = x^2 and y = 2x + 3 is 9 square units.
Explanation:
To find the area of the region enclosed between the curves y = x^2 and y = 2x + 3, we first need to determine the points of intersection between the two curves.
Setting the two equations equal to each other, we have:
x^2 = 2x + 3
Bringing all terms to one side, we get:
x^2 - 2x - 3 = 0
Factoring the quadratic equation, we have:
(x - 3)(x + 1) = 0
So, x = 3 or x = -1.
Now, we can integrate the difference between the two functions over the interval of intersection to find the area:
A = ∫(2x + 3 - x^2) dx
Integrating, we get:
A = [x^2 + 3x - (x^3/3)]
Substituting the limits of integration, we have:
A = [(3^2 + 3(3) - (3^3/3)) - ((-1)^2 + 3(-1) - ((-1)^3/3))]
Simplifying the expression, we get:
A = [9 + 9 - 9 - (1 + (-3) - (-1/3))]
A = [9]
Therefore, the area of the region enclosed between the curves y = x^2 and y = 2x + 3 is 9 square units.
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