Final answer:
The area of the region bounded by the curves y = x² and y = √ x is 1/3 square units, found by integrating the curves between their intersection points at x=0 and x=1 and taking the absolute value of the difference in their integrals.
Step-by-step explanation:
To find the area of the region bounded by the curves y = x² and y = √ x, we must first identify the points where the curves intersect. This is done by setting the equations equal to each other and solving for x:
x² = √ x ⇒ x² = x^(1/2) ⇒ x^(4/2) = x^(1/2) ⇒ x^(4 - 1/2) = 1 ⇒ x^(3/2) = 1 ⇒ x = 1
Now we know that the curves intersect at x=0 and x=1. To determine the area between them from x = 0 to x = 1, we subtract the integral of the lower curve from the integral of the upper curve:
∫ (x² - √ x) dx from x = 0 to x = 1
Computing the integrals yields the area:
∫ x² dx from x = 0 to x = 1 = [x^(3/3)] from 0 to 1 = (1/3) - 0 = 1/3
∫ √ x dx from x = 0 to x = 1 = [2/3 x^(3/2)] from 0 to 1 = (2/3)(1) - 0 = 2/3
Therefore, the area of the region bounded by the curves is:
Area = ∫ x² dx - ∫ √ x dx = (1/3) - (2/3) = -1/3
However, area cannot be negative. Since we subtracted the larger area (under the square root curve) from the smaller area (under the parabola), and both curves are above the x-axis, we should take the absolute value of our answer:
Area = |(1/3) - (2/3)| = 1/3