Area common to the curves `y=sqrt(x)` and `x=sqrt(y)` is (A) `1` (B) `2/3` (C) `1/3` (D) none of these

Options

A. 1
B. `2//3`
C. `1//3`
D. `4//3`

Answer

To find the area common to the curves \( y = \sqrt{x} \) and \( x = \sqrt{y} \), we can follow these steps:

### Step 1: Rewrite the equations
The first curve is given as \( y = \sqrt{x} \). Squaring both sides gives us:
\[ y^2 = x \quad \text{(Equation 1)} \]

The second curve is given as \( x = \sqrt{y} \). Squaring both sides gives us:
\[ x^2 = y \quad \text{(Equation 2)} \]

### Step 2: Find points of intersection
To find the points where these two curves intersect, we can set \( y \) from Equation 1 equal to \( y \) from Equation 2:
\[ \sqrt{x} = x^2 \]
Squaring both sides, we get:
\[ x = x^4 \]
Rearranging gives us:
\[ x^4 - x = 0 \]
Factoring out \( x \):
\[ x(x^3 - 1) = 0 \]
This gives us the solutions:
\[ x = 0 \quad \text{or} \quad x^3 = 1 \implies x = 1 \]
Thus, the points of intersection are \( (0, 0) \) and \( (1, 1) \).

### Step 3: Set up the integral for the area
The area between the curves from \( x = 0 \) to \( x = 1 \) can be found by integrating the difference between the upper curve and the lower curve. Here, \( y = \sqrt{x} \) is the upper curve and \( y = x^2 \) is the lower curve.

The area \( A \) can be expressed as:
\[ A = \int_{0}^{1} \left( \sqrt{x} - x^2 \right) \, dx \]

### Step 4: Compute the integral
Now we compute the integral:
\[ A = \int_{0}^{1} \sqrt{x} \, dx - \int_{0}^{1} x^2 \, dx \]

Calculating the first integral:
\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
Evaluating from 0 to 1:
\[ \left[ \frac{2}{3} x^{3/2} \right]_{0}^{1} = \frac{2}{3} (1^{3/2} - 0^{3/2}) = \frac{2}{3} \]

Calculating the second integral:
\[ \int x^2 \, dx = \frac{x^3}{3} \]
Evaluating from 0 to 1:
\[ \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]

### Step 5: Combine the results
Now we can combine the results from the two integrals:
\[ A = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \]

### Final Answer
Thus, the area common to the curves \( y = \sqrt{x} \) and \( x = \sqrt{y} \) is:
\[ \boxed{\frac{1}{3}} \]