The area between `x=y^2`and `x = 4`is divided into two equal parts by the line`x = a`, find the value of a.

Options

A. `a=7^(2/3)`
B. `a=2^(2/3)`
C. `a=5^(2/3)`
D. `a=4^(2/3)`

Explanation

To find the value of \( a \) that divides the area between the curves \( x = y^2 \) and \( x = 4 \) into two equal parts, we can follow these steps:

### Step 1: Determine the area between the curves
The area between the curves \( x = y^2 \) and \( x = 4 \) can be found by integrating the function from the leftmost intersection point to the rightmost intersection point.

First, we find the points of intersection:
\[ y^2 = 4 \implies y = \pm 2 \]
Thus, the area between the curves from \( y = -2 \) to \( y = 2 \) can be calculated using the integral:
\[ \text{Area} = \int_{-2}^{2} (4 - y^2) \, dy \]

### Step 2: Calculate the total area
Now we compute the integral:
\[ \text{Area} = \int_{-2}^{2} (4 - y^2) \, dy = \left[ 4y - \frac{y^3}{3} \right]_{-2}^{2} \]
Calculating the limits:
\[ = \left( 4(2) - \frac{(2)^3}{3} \right) - \left( 4(-2) - \frac{(-2)^3}{3} \right) \]
\[ = \left( 8 - \frac{8}{3} \right) - \left( -8 + \frac{8}{3} \right) \]
\[ = \left( 8 - \frac{8}{3} + 8 - \frac{8}{3} \right) \]
\[ = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3} \]

### Step 3: Set up the equation for equal areas
Since we want to divide this area into two equal parts, each part should have an area of:
\[ \frac{32}{3} \div 2 = \frac{16}{3} \]

### Step 4: Set up the integral for the area from \( 0 \) to \( a \)
The area from \( y = 0 \) to \( y = \sqrt{a} \) (where \( x = a \)) is:
\[ \text{Area}_{0 \text{ to } a} = \int_{0}^{\sqrt{a}} (4 - y^2) \, dy \]

Calculating this integral:
\[ = \left[ 4y - \frac{y^3}{3} \right]_{0}^{\sqrt{a}} = \left( 4\sqrt{a} - \frac{(\sqrt{a})^3}{3} \right) - (0) \]
\[ = 4\sqrt{a} - \frac{a\sqrt{a}}{3} \]

### Step 5: Set the area equal to \( \frac{16}{3} \)
Setting this equal to \( \frac{16}{3} \):
\[ 4\sqrt{a} - \frac{a\sqrt{a}}{3} = \frac{16}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 12\sqrt{a} - a\sqrt{a} = 16 \]
Rearranging gives:
\[ a\sqrt{a} - 12\sqrt{a} + 16 = 0 \]

### Step 6: Let \( u = \sqrt{a} \)
Substituting \( u = \sqrt{a} \):
\[ au - 12u + 16 = 0 \implies u^3 - 12u + 16 = 0 \]

### Step 7: Solve the cubic equation
Using trial and error or synthetic division, we can find that \( u = 2 \) is a root:
\[ (2)^3 - 12(2) + 16 = 0 \]
Thus, \( \sqrt{a} = 2 \) implies \( a = 4 \).

### Final Answer
The value of \( a \) that divides the area into two equal parts is:
\[ \boxed{4} \]