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 will follow these steps:

### Step 1: Determine the area between the curves
The area between the curves \( x = y^2 \) and \( x = 4 \) can be computed by integrating the difference of the functions over the appropriate limits.

The curves intersect when \( y^2 = 4 \), which gives \( y = -2 \) and \( y = 2 \). Thus, the area \( A \) can be calculated as:
\[ A = \int_{-2}^{2} (4 - y^2) \, dy \]

### Step 2: Calculate the area
Now, we compute the integral:
\[ A = \int_{-2}^{2} (4 - y^2) \, dy = \left[ 4y - \frac{y^3}{3} \right]_{-2}^{2} \]
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: Find half of the area
Since we want to divide this area into two equal parts, we need to find:
\[ \frac{A}{2} = \frac{16}{3} \]

### Step 4: Set up the equation for area to the left of \( x = a \)
The area to the left of the line \( x = a \) is given by:
\[ A_1 = \int_{-y_a}^{y_a} (a - y^2) \, dy \]
where \( y_a = \sqrt{a} \) (the y-coordinate corresponding to \( x = a \)).

Thus,
\[ A_1 = \int_{-\sqrt{a}}^{\sqrt{a}} (a - y^2) \, dy \]
Calculating this integral:
\[ = \left[ ay - \frac{y^3}{3} \right]_{-\sqrt{a}}^{\sqrt{a}} = \left( a\sqrt{a} - \frac{(\sqrt{a})^3}{3} \right) - \left( -a\sqrt{a} + \frac{(-\sqrt{a})^3}{3} \right) \]
\[ = \left( a\sqrt{a} - \frac{a\sqrt{a}}{3} \right) - \left( -a\sqrt{a} - \frac{-a\sqrt{a}}{3} \right) \]
\[ = 2a\sqrt{a} - \frac{2a\sqrt{a}}{3} = \frac{6a\sqrt{a}}{3} - \frac{2a\sqrt{a}}{3} = \frac{4a\sqrt{a}}{3} \]

### Step 5: Set the area equal to half the total area
We set this equal to \( \frac{16}{3} \):
\[ \frac{4a\sqrt{a}}{3} = \frac{16}{3} \]
Multiplying through by 3:
\[ 4a\sqrt{a} = 16 \]
Dividing by 4:
\[ a\sqrt{a} = 4 \]

### Step 6: Solve for \( a \)
Let \( a^{3/2} = 4 \). Squaring both sides gives:
\[ a^3 = 16 \]
Taking the cube root:
\[ a = 16^{1/3} = 4^{2/3} \]

Thus, the value of \( a \) is:
\[ \boxed{4^{2/3}} \]