Area bounded by curve `y^(2)=x` and `x=4` is divided into `4` equal parts by the lines `x=a` and `y=b` then `a& b` is

Options

A. Area of each part `= (8)/(3)`
B. b = 0
C. `a = sqrt(2)`
D. `a = (16)^(1//3)`

Answer

To solve the problem of finding the values of \( a \) and \( b \) such that the area bounded by the curve \( y^2 = x \) and the line \( x = 4 \) is divided into 4 equal parts, we can follow these steps:

### Step 1: Determine the area bounded by the curve and the line
The curve \( y^2 = x \) can be rewritten as \( y = \sqrt{x} \) for the upper half and \( y = -\sqrt{x} \) for the lower half. The area we are interested in is between \( x = 0 \) and \( x = 4 \).

The area \( A \) can be calculated using the integral:
\[ A = \int_{0}^{4} \sqrt{x} \, dx + \int_{0}^{4} -\sqrt{x} \, dx \]
However, since the area under the curve is symmetric about the x-axis, we can calculate the area above the x-axis and double it:
\[ A = 2 \int_{0}^{4} \sqrt{x} \, dx \]

### Step 2: Calculate the integral
The integral of \( \sqrt{x} \) is:
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} \]
Now, we evaluate it from 0 to 4:
\[ A = 2 \left[ \frac{2}{3} x^{3/2} \right]_{0}^{4} = 2 \left[ \frac{2}{3} (4^{3/2}) - 0 \right] \]
Calculating \( 4^{3/2} \):
\[ 4^{3/2} = (2^2)^{3/2} = 2^3 = 8 \]
Thus, the area becomes:
\[ A = 2 \left[ \frac{2}{3} \cdot 8 \right] = 2 \cdot \frac{16}{3} = \frac{32}{3} \]

### Step 3: Divide the area into 4 equal parts
To find the area of each part, we divide the total area by 4:
\[ \text{Area of each part} = \frac{A}{4} = \frac{32/3}{4} = \frac{8}{3} \]

### Step 4: Determine the value of \( a \)
Let \( a \) be the x-coordinate where the area from \( x = 0 \) to \( x = a \) equals \( \frac{8}{3} \):
\[ \int_{0}^{a} \sqrt{x} \, dx = \frac{8}{3} \]
Calculating the integral:
\[ \frac{2}{3} a^{3/2} = \frac{8}{3} \]
Multiplying both sides by 3:
\[ 2 a^{3/2} = 8 \]
Dividing by 2:
\[ a^{3/2} = 4 \]
Taking both sides to the power of \( \frac{2}{3} \):
\[ a = 4^{2/3} = (2^2)^{2/3} = 2^{4/3} = 16^{1/3} \]

### Step 5: Determine the value of \( b \)
Since the area is symmetric about the x-axis, the line \( y = b \) that divides the area into equal parts must be the x-axis:
\[ b = 0 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 16^{1/3}, \quad b = 0 \]