Determine the area enclosed by the curves y = x^2 and y = 4.
Practice Questions
1 question
Q1
Determine the area enclosed by the curves y = x^2 and y = 4.
8/3
4
16/3
2
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.
Questions & Step-by-step Solutions
1 item
Q
Q: Determine the area enclosed by the curves y = x^2 and y = 4.
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.
Steps: 8
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].
