Throughout this assignment you will work with the vector field H(x, y) = (0, x).

Question

Throughout this assignment you will work with the vector field H(x, y) = (0, x). To begin, let L be the line segment in the plane from (x_1, y_1) to (x_2, y_2). Give a parametric representation of L. Compute integral_L H Tds. Express your answer in terms of X_1, y_1, X_2, and y_2. Explain how the result indicates that: The integral of (0, X) over the line segment from (X_1, y_1) to (X_2, y_2) is the average of x at the ends of the segment times the change in y over the segment. Show that if we use this area-computation process to compute the area of the rectangle shown at the right, we obtain the usual area formula, A = LH. Show that if we use this area-computation process to compute the area of the triangle shown at the right, we obtain the usual area formula, A = LH/2.

Explanation / Answer

As per multiple question rules, we are supposed to answer the single question.

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