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To determine the limits for the integral, we need to understand the curve \( c \) and the path along which we are integrating. The problem specifies that the curve \( c \) is in the \( xy \)-plane and is given by the equation \( y = x^3 \). We are evaluating the line integral from the point \((1, 1)\) to the point \((2, 8)\).

Let's examine why the limits are from 1 to 2:

1. **Curve Description**: The curve \( c \) is described by \( y = x^3 \). This implies that \( y \) is a function of \( x \).

2. **Starting and Ending Points**: The integration is along the curve \( y = x^3 \) from the point \((1, 1)\) to the point \((2, 8)\).
   - At the starting point \((1, 1)\), \( x = 1 \) and \( y = 1^3 = 1 \).
   - At the ending point \((2, 8)\), \( x = 2 \) and \( y = 2^3 = 8 \).

Since \( y \) is expressed in terms of \( x \) and the points are given in terms of \( x \)-coordinates ranging from 1 to 2, the limits of integration for \( x \) naturally range from 1 to 2.

Therefore, the integral limits are from \( x = 1 \) to \( x = 2 \), corresponding to the \( x \)-coordinates of the given points on the curve \( y = x^3 \).

Do you need further details or have any questions?

Here are some questions you might want to ask:
1. How is the vector field \(\mathbf{F}\) expressed in terms of \( x \) and \( y \)?
2. How do we convert the differential line element \( d\mathbf{r} \) in terms of \( dx \) and \( dy \)?
3. What are the steps to integrate the given functions over the specified limits?
4. How do we verify the result of the integral calculation?
5. Can you explain the concept of a line integral in the context of vector fields?

**Tip:** When performing line integrals, ensure that the limits of integration correspond to the parameter used to describe the path. In this case, \( x \) varies from 1 to 2 along the curve \( y = x^3 \).

Explore the concept of line integrals and understand how limits of integration are determined. This example focuses on integrating along a curve described by y = x^3 from (1, 1) to (2, 8). Learn the mathematical reasoning behind choosing integration limits based on the curve's parameterization.

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