The changes in a function y and the independent variable x are related as `dy/dx=x^2`. Find y as a function of x.

Explanation

To solve the problem where the changes in a function \( y \) and the independent variable \( x \) are related as \( \frac{dy}{dx} = x^2 \), we will follow these steps:

### Step 1: Write the differential equation
We start with the given relationship:
\[ \frac{dy}{dx} = x^2 \]

### Step 2: Rearrange the equation
We can rearrange this equation to separate the variables \( y \) and \( x \):
\[ dy = x^2 \, dx \]

### Step 3: Integrate both sides
Next, we integrate both sides of the equation. The left side will be integrated with respect to \( y \) and the right side with respect to \( x \):
\[ \int dy = \int x^2 \, dx \]

### Step 4: Perform the integration
The integral of \( dy \) is simply \( y \). For the right side, the integral of \( x^2 \) is:
\[ \int x^2 \, dx = \frac{x^3}{3} + C \]
where \( C \) is the constant of integration.

### Step 5: Write the final expression for \( y \)
Combining the results from the integration, we have:
\[ y = \frac{x^3}{3} + C \]

### Final Result
Thus, the function \( y \) as a function of \( x \) is:
\[ y = \frac{x^3}{3} + C \]

---