12. 1111111 Carefully explain, using diagrams and definitions as needed, what a scalar line integral represents.

Q: Find L{f (t)}using the First Shifting Theorem, Note: a, b, e, w, k and n are constants 1. Il…

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Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: State what it means for a scalar function y = f (x) to beintegrable on an interval [a, b].

A: Scalar integrable  function y = f (x)

Q: What is the graph of r(t)=2 costi + sintj?

A: Vector-valued function : It is a function of the form  r(t)=f(t)i⏜+g(t)j⏜ or…

Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: Use Green's Theorem to evaluate the following integral Let² dx + (5x + 9) dy Where C is the triangle…

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Q: Use Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise.…

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Q: calculate div(F) and curl(F). F = (xy, yz, y² – x³)

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Q: Use the Fundamental Theorem of Calculus to find the area under curve of c)= 6z 4 13 between I=14 and…

A: Given function: f(x)=6x+13 Definition: Fundamental theorem of calculus: It states that f is…

Q: What is the geometrical meaning of an integral of a vector function?

A: We can realize geometric meaning of integration such as

Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: V × (V × F) = V(V-F) – V°F. %3D

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Q: Determine whether the line integral of each vector field (in blue) along the semicircular, oriented…

A: Determine whether the line integral of each vector field along the semicircular or oriented path is…

Q: Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented…

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Q: O (x² – y²) dx + x dy where is the circle x + y² = 9.

A: Green's theorem states that ∮CM dx+N dy=∬R∂M∂y-∂M∂ydx dy Here M=x2-y2 and N=x Hence the integral…

Q: Evaluate the circulation of G = ayi+zj+ 7yk around a square of side 9, centered at the origin, lying…

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Q: grad (фu) —D и @ grad ф + ф grad u

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Q: Let F = yi + xj + x³y³ z³k, C the curve x2 + y² + 6x – 2y + 9 = 0, z = 2 (with counterclockwise…

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Q: Evaluate the integral of the vector function

A: To find the integration

Q: Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented…

A: Greens theorem

Q: Evaluate 6z ds where Cis the curve y = -72² from (-1,-7) to (1, -7). Enter the scalar line integral…

A: To evaluate ∫C6xds where C is the curve y=−7x2 from −1,−7 to 1,−7.