In this lesson, we will learn how to find the line integral of a vector field along a curve with an orientation.

Students will be able to

Q1:

Suppose 𝐶 is the path given by r(𝑡)=(𝑡,𝑡) for 0≤𝑡≤1, 𝐶 is the path given by r(𝑡)=(1−𝑡,1−𝑡) for 0≤𝑡≤1, and Fij=𝑥+(𝑦+1)ln. Without calculating the integrals, which of the following is true?

Q2:

In the figure, the curve 𝐶 from 𝑃 to 𝑄 consists of two quarter-unit circles, one with center (1, 0) and the other with center (3, 0). Calculate the line integral ⋅Frd, where Fij=𝑥2𝑦2−𝑥2𝑦2sinsincoscos.

Q3:

Let 𝑃 be the arc of a unit circle in the 𝑥𝑦-plane traversed counterclockwise from (0,1) to (1,0). Determine the exact value of the line integral of the vector field Fijk(𝑥,𝑦,𝑧)=3𝑥𝑒+2𝑦𝑧𝑒+𝑦𝑒 over 𝑃.

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