To evaluate the line integral of the vector field $\mathbf{F}(x, y) = \langle x^2, y^2 \rangle$ along a parabolic path from $(0, 0)$ to $(1, 1)$, let’s proceed step by step.

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1. Represent the Path

We need a parabolic path. A common choice is: $y = x^2$ with $x$ varying from $0$ to $1$. Therefore, the path can be parameterized as: $\mathbf{r}(t) = \langle t, t^2 \rangle, \quad t \in [0, 1].$

The differential of $\mathbf{r}(t)$ is: $\mathbf{r}'(t) = \frac{d}{dt} \langle t, t^2 \rangle = \langle 1, 2t \rangle,$ and thus: $d\mathbf{r} = \mathbf{r}'(t) \, dt = \langle 1, 2t \rangle \, dt.$

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2. Write the Line Integral

The line integral is defined as: $\int_C \mathbf{F} \cdot d\mathbf{r},$ where $\mathbf{F} = \langle x^2, y^2 \rangle$. Substituting the parameterization $(x, y) = (t, t^2)$: $\mathbf{F}(t) = \langle t^2, (t^2)^2 \rangle = \langle t^2, t^4 \rangle.$

Thus, the dot product $\mathbf{F} \cdot d\mathbf{r}$ becomes: $\mathbf{F} \cdot d\mathbf{r} = \langle t^2, t^4 \rangle \cdot \langle 1, 2t \rangle = t^2(1) + t^4(2t) = t^2 + 2t^5.$

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3. Evaluate the Integral

Now, integrate $t^2 + 2t^5$ over $t \in [0, 1]$: $\int_0^1 (t^2 + 2t^5) \, dt = \int_0^1 t^2 \, dt + \int_0^1 2t^5 \, dt.$

For the first term: $\int_0^1 t^2 \, dt = \left[\frac{t^3}{3}\right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}.$

For the second term: $\int_0^1 2t^5 \, dt = 2 \int_0^1 t^5 \, dt = 2 \left[\frac{t^6}{6}\right]_0^1 = 2 \left(\frac{1^6}{6} - \frac{0^6}{6}\right) = \frac{2}{6} = \frac{1}{3}.$

Add the two results: $\int_0^1 (t^2 + 2t^5) \, dt = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}.$

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Final Answer:

The value of the line integral is: $\boxed{\frac{2}{3}}$

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Do you want detailed steps for each parameterization, or have any related questions? Here are 5 follow-up questions to deepen your understanding:

1. How does the choice of path affect the line integral in vector fields?
2. Can you explain the geometric meaning of the line integral in this context?
3. What happens if you use a different parabolic path, such as $y = x^3$?
4. How would the integral change if $\mathbf{F}(x, y)$ had an additional term, like $\langle x^2, y^2 + x \rangle$?
5. What is the relationship between this integral and the concept of work done in physics?

Tip: Always ensure your parameterization matches the given path before computing line integrals!