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How to Integrate line resolution

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Professor Dave here, I want to tell you about line integrals. By now we are very familiar with ordinary integrals. Integrating f(x)dx lets us find the area under a curve given by the function f(x). Integrating f(x,y)dxdy lets us find the volume under a surface given by the function f(x,y). Now with line integrals, we can integrate a surface f(x,y) along the path of some curve C. We will be integrating along small segments of the curve C, which we will call ds. Just like when we initially learned integration, these segments will be our widths, while the surface f(x,y) will be our heights. So once again, our integration will give an area, but now we are finding the area under a surface along a particular path within that surface. The way we will write this out is the integration along C of f(x,y)ds. Recall that when we learned how to evaluate multiple integrals, the x and y variables were treated independently during the integration. But now with line integrals we have a single integral

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How do you know if an integral is positive or negative?

If ALL of the area within the interval exists above the x-axis yet below the curve then the result is positive . If ALL of the area within the interval exists below the x-axis yet above the curve then the result is negative .

Is a vector field positive or negative?

If the vector field is increasing in magnitude as you move along the flow of a vector field, then the divergence is positive. If the vector field is decreasing in magnitude as you move along the flow of a vector field, then the divergence is negative.

What does a line integral tell us?

A line integral is used to calculate the mass of wire. It helps to calculate the moment of inertia and centre of mass of wire. It is used in Amperes Law to compute the magnetic field around a conductor. In Faradays Law of Magnetic Induction, a line integral helps to determine the voltage generated in a loop.

Is line integral always positive?

Yes, positive when they are in the same direction, negative when they are opposite.

What is the integral of a straight line?

In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve.

What does a positive line integral mean?

The line integral essentially measures the extent to which the path and vector field are. aligned. The line integral will have a positive sign if the path and field are oriented in the same direction, and will be negative if they are in opposite directions.

What does the integral of a line tell you?

A line integral is used to calculate the surface area in the three-dimensional planes. Some of the applications of line integrals in the vector calculus are as follows: A line integral is used to calculate the mass of wire. It helps to calculate the moment of inertia and centre of mass of wire.

What is a line integral conceptually?

A line integral (sometimes called a path integral) is the integral of some function along a curve. One can integrate a scalar-valued function along a curve, obtaining for example, the mass of a wire from its density. One can also integrate a certain type of vector-valued functions along a curve.

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