To compute the line integral over C, we first need to parameterize the curve C. The curve is given by the parabola x = y^2, and we are moving from the point (1, -1) to the point (1, 1). A suitable parameterization is then given by r(t) = (t^2, t), where t ranges from -1 to 1.

The line integral of a function f over a curve C parameterized by r(t), a ≤ t ≤ b, is given by ∫_C f ds = ∫_a^b f(r(t)) ||r'(t)|| dt.

First, we need to compute the derivative r'(t). We have r'(t) = (2t, 1).

The magnitude of r'(t) is then ||r'(t)|| = sqrt((2t)^2 + 1^2) = sqrt(4t^2 + 1).

We then need to compute the function f at the points on the curve, i.e., f(r(t)). However, you have not specified the function f in your question. If you provide the function, I can continue with the computation.

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To compute the line integral over C, we first need to parameterize the curve C. The curve is given by the parabola x = y^2, and we are moving from the point (1, -1) to the point (1, 1). A suitable parameterization is then given by r(t) = (t^2, t), where t ranges from -1 to 1.

The line integral of a function f over a curve C parameterized by r(t), a ≤ t ≤ b, is given by ∫_C f ds = ∫_a^b f(r(t)) ||r'(t)|| dt.

First, we need to compute the derivative r'(t). We have r'(t) = (2t, 1).

The magnitude of r'(t) is then ||r'(t)|| = sqrt((2t)^2 + 1^2) = sqrt(4t^2 + 1).

We then need to compute the function f at the points on the curve, i.e., f(r(t)). However, you have not specified the function f in your question. If you provide the function, I can continue with the computation.

Knowee AI · Accepted Answer

To compute the line integral over C, we first need to parameterize the curve C. The curve is given by the parabola x = y^2, and we are moving from the point (1, -1) to the point (1, 1). A suitable parameterization is then given by r(t) = (t^2, t), where t ranges from -1 to 1.

The line integral of a function f over a curve C parameterized by r(t), a ≤ t ≤ b, is given by ∫_C f ds = ∫_a^b f(r(t)) ||r'(t)|| dt.

First, we need to compute the derivative r'(t). We have r'(t) = (2t, 1).

The magnitude of r'(t) is then ||r'(t)|| = sqrt((2t)^2 + 1^2) = sqrt(4t^2 + 1).

We then need to compute the function f at the points on the curve, i.e., f(r(t)). However, you have not specified the function f in your question. If you provide the function, I can continue with the computation.

The line integral over C, the arc of the parabola x = y2 from (1,-1) to (1,1)

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