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We are asked to evaluate the line integral:

\[
\int_{\Gamma} x \, ds
\]

where \(\Gamma\) is the line segment joining the points \((0, 1)\) and \((1, 0)\) on the \(xy\)-plane.

### Step 1: Parametrize the Line Segment
The line segment \(\Gamma\) can be parametrized as:
\[
\mathbf{r}(t) = (1 - t)(0, 1) + t(1, 0) = (t, 1 - t), \quad t \in [0, 1]
\]
So, \(x(t) = t\) and \(y(t) = 1 - t\).

### Step 2: Compute \(ds\)
The differential arc length element \(ds\) is given by:
\[
ds = \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt
\]
First, compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\):
\[
\frac{dx}{dt} = 1, \quad \frac{dy}{dt} = -1
\]
Now compute \(ds\):
\[
ds = \sqrt{(1)^2 + (-1)^2} \, dt = \sqrt{2} \, dt
\]

### Step 3: Set Up the Integral
Now, substitute \(x = t\) and \(ds = \sqrt{2} \, dt\) into the integral:
\[
\int_{\Gamma} x \, ds = \int_0^1 t \cdot \sqrt{2} \, dt
\]

### Step 4: Evaluate the Integral
\[
\int_0^1 t \cdot \sqrt{2} \, dt = \sqrt{2} \int_0^1 t \, dt = \sqrt{2} \left[ \frac{t^2}{2} \right]_0^1 = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}
\]

### Final Answer:
The value of the line integral is \(\frac{\sqrt{2}}{2}\), which corresponds to option **B**.

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Do you want more details, or do you have any other questions?

Here are 5 related questions:
1. How would the integral change if the function was \(y \cdot ds\) instead of \(x \cdot ds\)?
2. What happens if we change the curve \(\Gamma\) to a circular arc instead of a straight line?
3. How do you compute the line integral of a vector field over a curve?
4. How does this process generalize to higher dimensions (e.g., surface integrals)?
5. What role does the parametrization of the curve play in evaluating line integrals?

**Tip**: Always double-check your parametrization and the limits of integration, as they are crucial for computing line integrals correctly.

Evaluate the line integral where Γ is the line segment joining (0,1) and (1,0) on the xy-plane.

Learn how to evaluate the line integral ∫Γ x ds where Γ is the line segment between (0,1) and (1,0) on the xy-plane. This problem covers concepts such as curve parametrization, arc length differentials, and line integrals of scalar functions. Suitable for advanced calculus students.

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