Evaluation of line integrals over piecewise smooth curves is a relatively simple thing to do. All we do is evaluate the line integral over each of the pieces and then add them up. The line integral for some function over the above piecewise curve would be,. Notice that we put direction arrows on the curve in the above example. The direction of motion along a curve may change the value of the line integral as we will see in the next section. Cubing it out is not that difficult, but it is more work than a simple substitution.
So, the previous two examples seem to suggest that if we change the path between two points then the value of the line integral with respect to arc length will change.
In a later section we will investigate this idea in more detail. Here is the parameterization of the curve. So, it looks like when we switch the direction of the curve the line integral with respect to arc length will not change. This will always be true for these kinds of line integrals.
So, for a line integral with respect to arc length we can change the direction of the curve and not change the value of the integral. This is a useful fact to remember as some line integrals will be easier in one direction than the other.
There are two parameterizations that we could use here for this curve. The first is to use the formula we used in the previous couple of examples. That parameterization is,. Sometimes we have no choice but to use this parameterization. However, in this case there is a second probably easier parameterization. Using this the parameterization is,. This will be a much easier parameterization to use so we will use this. Here is the line integral for this curve.
Note that this time, unlike the line integral we worked with in Examples 2, 3, and 4 we got the same value for the integral despite the fact that the path is different. This will happen on occasion. We should also not expect this integral to be the same for all paths between these two points.
At this point all we know is that for these two paths the line integral will have the same value. It is completely possible that there is another path between these two points that will give a different value for the line integral.
The fact tells us that this line integral should be the same as the second part i. However, there is no reason to restrict ourselves like that. We can do line integrals over three-dimensional curves as well.
Note that often when dealing with three-dimensional space the parameterization will be given as a vector function. Notice that we changed up the notation for the parameterization a little. Then as we have seen in section Example Now we turn to a perhaps more interesting example.
Recall that in the simplest case, the work done by a force on an object is equal to the magnitude of the force times the distance the object moves; this assumes that the force is constant and in the direction of motion.
We have already dealt with examples in which the force is not constant; now we are prepared to examine what happens when the force is not parallel to the direction of motion. We have already examined the idea of components of force, in example In other words, work is computed using a particular line integral of the form we have considered.
Find the work done. Ex Home » Vector Calculus » Line Integrals. Collapse menu Introduction 1 Analytic Geometry 1. Lines 2. Distance Between Two Points; Circles 3. Functions 4. The slope of a function 2. An example 3. Limits 4. The Derivative Function 5. Properties of Functions 3 Rules for Finding Derivatives 1. The Power Rule 2. Linearity of the Derivative 3. The Product Rule 4. The Quotient Rule 5.
The Chain Rule 4 Transcendental Functions 1. Trigonometric Functions 2. A hard limit 4. Derivatives of the Trigonometric Functions 6. Exponential and Logarithmic functions 7. Derivatives of the exponential and logarithmic functions 8. Implicit Differentiation 9. Inverse Trigonometric Functions Limits revisited Hyperbolic Functions 5 Curve Sketching 1. Maxima and Minima 2. The first derivative test 3. The second derivative test 4. Concavity and inflection points 5. Optimization 2.
Related Rates 3. Newton's Method 4. Linear Approximations 5. The Mean Value Theorem 7 Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3. Some Properties of Integrals 8 Techniques of Integration 1.
Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4. Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Additional exercises 9 Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4.
Parametric Equations 5.
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