The area of the region bounded by the curve `x = ay^(2)` and `y = 1` is equal to

Options

A. `1-(2)/(3(sqrta))`
B. `(a)/(3)`
C. `(a)/(2)`
D. 3

Answer

To find the area of the region bounded by the curve \( x = ay^2 \) and the line \( y = 1 \), we will follow these steps:

### Step 1: Understand the curves
The curve \( x = ay^2 \) is a parabola that opens to the right, and the line \( y = 1 \) is a horizontal line. We need to find the area between these two curves from the point where they intersect.

### Step 2: Find the points of intersection
To find the points of intersection, we set \( y = 1 \) in the equation of the curve:
\[ x = a(1)^2 = a \]
Thus, the curves intersect at the point \( (a, 1) \) and also at the origin \( (0, 0) \).

### Step 3: Set up the integral for the area
The area \( A \) between the curves can be calculated by integrating the difference between the right curve and the left curve. In this case, the left curve is the y-axis (where \( x = 0 \)) and the right curve is \( x = ay^2 \). The area can be expressed as:
\[ A = \int_{y=0}^{y=1} (ay^2 - 0) \, dy \]

### Step 4: Evaluate the integral
Now we compute the integral:
\[ A = \int_{0}^{1} ay^2 \, dy \]
Calculating this integral:
\[ A = a \int_{0}^{1} y^2 \, dy \]
The integral of \( y^2 \) is:
\[ \int y^2 \, dy = \frac{y^3}{3} \]
So we have:
\[ A = a \left[ \frac{y^3}{3} \right]_{0}^{1} = a \left( \frac{1^3}{3} - \frac{0^3}{3} \right) = a \cdot \frac{1}{3} = \frac{a}{3} \]

### Final Answer
Thus, the area of the region bounded by the curve \( x = ay^2 \) and the line \( y = 1 \) is:
\[ \frac{a}{3} \text{ square units} \]

---