find the area of the shaded region y=x , y=x^2 – Free 80A

Understanding the Curves:
We are analyzing the region bounded between the line y = x and the parabola y = x². To calculate the area of the shaded region between these two curves, we integrate the vertical distance between them across their intersection interval.

Step 1: Find the Points of Intersection

To find the points where the curves intersect, set:

x = x²
⇒ x² – x = 0
⇒ x(x – 1) = 0
⇒ x = 0 and x = 1

So, the two curves intersect at x = 0 and x = 1.

Step 2: Set Up the Definite Integral

From x = 0 to x = 1, the line y = x lies above the parabola y = x². So, the vertical distance between them is:

Area = ∫ from 0 to 1 of (x – x²) dx

Step 3: Integrate the Expression

Now integrate the function (x – x²):

∫ (x – x²) dx = ∫ x dx – ∫ x² dx
= (1/2)x² – (1/3)x³

Evaluate from 0 to 1:

[(1/2)(1)² – (1/3)(1)³] – [(1/2)(0)² – (1/3)(0)³]
= (1/2 – 1/3) – 0 = 1/6

Final Answer:

The area of the shaded region bounded between the line y = x and the parabola y = x² from x = 0 to x = 1 is:

A = 1/6 square units

Visual Insight

The parabola opens upward and lies below the line y = x between the points x = 0 and x = 1. The region enclosed is a curved triangular shape, with the vertical distance between the curves shrinking to zero at both endpoints.

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