Can someone simply explain to me how to calculate linear integral linke below?∫L5ydLwhere L is line segment from (0;0) to (0,2;0,2)

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afadimoz4

2022-03-19

Can someone simply explain to me how to calculate linear integral linke below?

\int_{L} 5 y d L

where L is line segment from (0;0) to (0,2;0,2)

Porter Camacho

Beginner2022-03-20Added 6 answers

Assuming you mean to calculate the line integral

\int_{L} 5 y d s

, where L is the segment from (0,0) to (0.2,0.2), then define a bijective parametrization

r : [0, 0.2] \to L

r (t) = (t, t)

. Also let

f (x, y) = 5 y

. Then,

\int_{L} f, d s = \int_{0}^{0.2} f (r (t)) | r^{'} (t) |, d t

Here,

f (r (t)) = f (t, t) = 5 t

and

| r^{'} (t) | = | \begin{array}{cc} 1 & 1 \end{array} | = \sqrt{2}

Thus,

\int_{L} 5 y, d s = \sqrt{2} \int_{0}^{0.2} 5 t, d t

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