Question

Evaluate the line integral integral (y² dx - 2x² dy) along the parabola y = x² from the point (0, 0) to the point (2, 4).

Answer 1

To evaluate the line integral along the parabola y = x² from (0, 0) to (2, 4), we can parameterize the parabola in terms of x and substitute the values into the line integral. Evaluating the integral will give us the desired value.

Step-by-step explanation:

To evaluate the line integral ∫(y² dx - 2x² dy) along the parabola y = x² from the point (0, 0) to the point (2, 4), we can first parameterize the parabola in terms of x. Since y = x², we can write x = t and y = t², where t varies from 0 to 2.

Next, we substitute these values into the line integral and evaluate it over the given limits of t.

The line integral becomes ∫(t^4 dt - 2t^2(2t dt)), where the limits of integration are from 0 to 2. Evaluating this integral, we get the result.