Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.

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Question: Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.

Options:

Correct Answer: 4

Solution:

The area between the curves y = x^2 and y = 4 is given by ∫(from 0 to 2) (4 - x^2) dx = [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 4/3.

Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.

Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.

Correct Answer: 4/3

Step 1: Identify the curves. We have two curves: y = x^2 (a parabola) and y = 4 (a horizontal line).
Step 2: Determine the points of intersection. Set x^2 = 4 to find where the curves meet. Solve for x: x = 2 and x = -2. We only need x = 0 to x = 2.
Step 3: Set up the integral to find the area between the curves. The area A is given by the integral from 0 to 2 of (top curve - bottom curve). Here, the top curve is y = 4 and the bottom curve is y = x^2.
Step 4: Write the integral: A = ∫(from 0 to 2) (4 - x^2) dx.
Step 5: Calculate the integral. First, find the antiderivative of (4 - x^2), which is 4x - (x^3)/3.
Step 6: Evaluate the integral from 0 to 2. Substitute x = 2 into the antiderivative: 4(2) - (2^3)/3 = 8 - 8/3.
Step 7: Simplify the result. 8 can be written as 24/3, so 24/3 - 8/3 = 16/3.
Step 8: The area between the curves from x = 0 to x = 2 is 16/3.

Area Between Curves – The process of finding the area between two curves involves integrating the difference of the functions over a specified interval.
Definite Integrals – Understanding how to evaluate definite integrals is crucial for calculating the area under curves.
Function Intersection – Identifying the points where the curves intersect helps in determining the limits of integration.