Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
Practice Questions
1 question
Q1
Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
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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.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
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.
Steps: 8
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.
