Question

Show that the value of \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) is the same for each parametric representation of \(C\). \(\mathbf{F}(x, y)=y \mathbf{i}+x^{2} \mathbf{j}\) (a) \(\mathbf{r}_{1}(t)=(2+t) \mathbf{i}+(3-t) \mathbf{j}, \quad 0 \leq t \leq 3\) (b) \(\mathbf{r}_{2}(w)=(2+\ln w) \mathbf{i}+(3-\ln w) \mathbf{j}, \quad 1 \leq w \leq e^{3}\)

Short answer

The obtained values for the line integral ∫C F · dr using the two parametric representations \(r_1(t)\) and \(r_2(w)\) are the same, thus the statement is proven.

Step-by-step solution

Compute the line integral using the first parametric representation

First, parametrize \(F\) using \(r_1(t)\) to obtain \(F_1(t)\) and calculate the derivative of \(r_1(t)\) which we'll call \(r_1'(t)\). Then calculate the dot product between these two vector functions and integrate it over the interval [0,3].

Compute the line integral using the second parametric representation

Similarly, parametrize \(F\) using \(r_2(w)\) to obtain \(F_2(w)\), calculate derivative of \(r_2(w)\) which we'll be calling \(r_2'(w)\). Proceed to take the dot product and integrate it over the interval [1, e^3].

Comparison of results

Finally, the calculated results from step 1 and step 2 will be compared. If they're the same, this verifies that the value of ∫C F · dr is the same for both parametric representations of C.