Let r be the region bounded by the graphs of y = x and y = x^2. How do you find the area of r?

To find the area of the region r bounded by the curves y = x and y = x², we first need to determine the points where these two curves intersect.

Setting the equations equal to each other:

x = x²

Rearranging gives:

x² – x = 0

Factoring out the common term:

x(x – 1) = 0

This gives us the solutions: x = 0 and x = 1. These points represent the x-coordinates of the intersection of the two curves.

Next, we can visualize the region r. The curve y = x is a straight line, and it is above the curve y = x² between the points of intersection (from x = 0 to x = 1).

The area A of the region can be found using the integral of the upper curve minus the lower curve:

A = ∫₀¹ (x – x²) dx

Now we can compute this integral:

A = ∫₀¹ (x – x²) dx

Calculating the integral gives:

A = ∫₀¹ x dx – ∫₀¹ x² dx

This results in:

A = [1/2 * x²] ₀¹ – [1/3 * x³] ₀¹

Evaluating each term:

A = (1/2 * 1² – 1/2 * 0²) – (1/3 * 1³ – 1/3 * 0³)

This simplifies to:

A = (1/2) – (1/3)

To combine these fractions, we find a common denominator (which is 6):

A = (3/6) – (2/6) = 1/6

Thus, the area of the region r bounded by the graphs of y = x and y = x² is 1/6.