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

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

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).
Step 7: Simplify the result: 8 - 8/3 = 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 – Calculating the area between two curves involves integrating the difference of the functions over a specified interval.
Definite integrals – Understanding how to evaluate definite integrals and apply the Fundamental Theorem of Calculus.
Function intersection – Identifying the points where the curves intersect to determine the limits of integration.