Determine the area enclosed by the curves y = x^2 and y = 4.

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Question: Determine the area enclosed by the curves y = x^2 and y = 4.

Options:

Correct Answer: 16/3

Solution:

The area enclosed is found by integrating from -2 to 2: ∫(from -2 to 2) (4 - x^2) dx = [4x - x^3/3] from -2 to 2 = (8 - 8/3) - (-8 + 8/3) = 16/3.

Determine the area enclosed by the curves y = x^2 and y = 4.

Determine the area enclosed by the curves y = x^2 and y = 4.

Correct Answer: 16/3

Step 1: Identify the curves. We have y = x^2 (a parabola) and y = 4 (a horizontal line).
Step 2: Find the points where the curves intersect. Set x^2 = 4. This gives x = -2 and x = 2.
Step 3: Determine the area between the curves from x = -2 to x = 2. The area is found by integrating the difference between the top curve (y = 4) and the bottom curve (y = x^2).
Step 4: Set up the integral: Area = ∫(from -2 to 2) (4 - x^2) dx.
Step 5: Calculate the integral. The integral of (4 - x^2) is 4x - (x^3)/3.
Step 6: Evaluate the integral from -2 to 2: [4(2) - (2^3)/3] - [4(-2) - ((-2)^3)/3].
Step 7: Simplify the expression: (8 - 8/3) - (-8 + 8/3).
Step 8: Combine the results to find the total area: 16/3.

Area between curves – The question tests the ability to find the area enclosed by two curves by setting up and evaluating an integral.
Integration – The solution requires knowledge of definite integrals and the fundamental theorem of calculus.
Identifying bounds – Students must correctly identify the points of intersection of the curves to set the limits of integration.