The area between the curve `x=-2y^(2)and x=1-3y^(2),` is

Options

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

Answer

To find the area between the curves \( x = -2y^2 \) and \( x = 1 - 3y^2 \), we will follow these steps:

### Step 1: Find the Points of Intersection
To find the points where the curves intersect, we set the equations equal to each other:

\[ -2y^2 = 1 - 3y^2 \]

Rearranging gives:

\[ 3y^2 - 2y^2 = 1 \implies y^2 = 1 \implies y = \pm 1 \]

### Step 2: Determine the Corresponding x-values
Now, we substitute \( y = 1 \) and \( y = -1 \) back into either equation to find the corresponding x-values. Using \( x = -2y^2 \):

For \( y = 1 \):

\[ x = -2(1)^2 = -2 \]

For \( y = -1 \):

\[ x = -2(-1)^2 = -2 \]

Thus, the points of intersection are \( (-2, 1) \) and \( (-2, -1) \).

### Step 3: Set Up the Integral for Area
The area between the curves can be found using the integral:

\[ \text{Area} = \int_{-1}^{1} [(\text{outer curve}) - (\text{inner curve})] \, dy \]

Here, the outer curve is \( x = 1 - 3y^2 \) and the inner curve is \( x = -2y^2 \).

### Step 4: Write the Integral
Thus, the area can be expressed as:

\[ \text{Area} = \int_{-1}^{1} [(1 - 3y^2) - (-2y^2)] \, dy \]

This simplifies to:

\[ \text{Area} = \int_{-1}^{1} [1 - 3y^2 + 2y^2] \, dy = \int_{-1}^{1} [1 - y^2] \, dy \]

### Step 5: Evaluate the Integral
Now we evaluate the integral:

\[ \text{Area} = \int_{-1}^{1} (1 - y^2) \, dy \]

Calculating this integral:

\[ = \left[ y - \frac{y^3}{3} \right]_{-1}^{1} \]

Calculating the upper limit:

\[ = \left( 1 - \frac{1^3}{3} \right) - \left( -1 + \frac{(-1)^3}{3} \right) \]

\[ = \left( 1 - \frac{1}{3} \right) - \left( -1 - \frac{1}{3} \right) \]

\[ = \left( \frac{2}{3} \right) - \left( -\frac{4}{3} \right) = \frac{2}{3} + \frac{4}{3} = \frac{6}{3} = 2 \]

### Step 6: Final Area Calculation
Thus, the area between the curves is:

\[ \text{Area} = 2 \text{ square units} \]

### Summary of Steps
1. Find points of intersection by solving \( -2y^2 = 1 - 3y^2 \).
2. Determine corresponding x-values for the intersection points.
3. Set up the integral for the area between the curves.
4. Evaluate the integral to find the area.