Calculate the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
Practice Questions
1 question
Q1
Calculate the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
4
8
6
2
The area is given by the integral from 0 to 2 of (4 - x^2) dx. This evaluates to [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 16/3.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
Solution: The area is given by the integral from 0 to 2 of (4 - x^2) dx. This evaluates to [4x - x^3/3] from 0 to 2 = (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: Determine the points of intersection of the curves. Set x^2 = 4. This gives x = 2 and x = -2, but we only care about x = 0 to x = 2.
Step 3: Find the area between the curves from x = 0 to x = 2. The area is calculated by integrating the difference between the upper curve (y = 4) and the lower curve (y = x^2).
Step 4: Set up the integral: Area = ∫ from 0 to 2 of (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 0 to 2: Substitute x = 2 into (4x - (x^3)/3) to get (4*2 - (2^3)/3) = (8 - 8/3).
