Finding the Area between graphs EXAMPLE 1

The video introduces an example for finding the area between two graphs, specifically y = sqrt(x) and y = x^2, using integral calculus.

The first step is to draw the graphs to visualize the problem. y = x^2 is a parabola opening to the positive y-axis. y^2 = x (derived from y = sqrt(x)) is a parabola opening to the positive x-axis. The region bounded by these two graphs is identified.

To calculate the area, a vertical strip method is applied. The formula for the area using vertical strips is the integral of (y_high - y_low) dx.

The limits of integration are the x-values where the two graphs intersect. By setting the equations equal to each other (y = x^2 and x = y^2), the intersection points are found to be (0,0) and (1,1). Therefore, the limits for x are from 0 to 1.

The integral is set up with the limits from 0 to 1. y_high is sqrt(x) or x^(1/2), and y_low is x^2. The integral is from 0 to 1 of (x^(1/2) - x^2) dx. The solution to this integral is 1/3 square units.