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Unformatted text preview: r that So So our definite integral becomes: We don’t have to parameterize the curve with arc length! Example:
Integrate over the circular arc We first find This curve happens to be parameterized by arc length. Example:
Evaluate where C is the segment of the line y = 2x from ( -1 , -2 ) to ( 1 , 2 ). Parameterize C: Then Let’s change the orientation of the curve so we start at ( 1 , 2 ) and stop at ( -1 , -2 ).
Thus are parameterization for C: : Then NOTE: The line integral with respect to ARC LENGTH is the same no matter which orientation you use!
(this will not be the case for other line integrals coming up)
The Calculus of WORK
1 For a constant force applied along a line Work = Force X Distance 2 For a varying force f(x) applied along a line Work = 3 For a constant force applied at an angle to the Work =
line of motion 4 And now, For a varying force applied along a curve: Derivation:
represent a force (vector) field. Find the work done by
particle moving along a curve C parameterized by for a We know WORK = FORCE X DISTANCE where the force is constant and in the direction of the displacement. Remember
from chapter 10 we derived the dot product definition when the force was constant and not in the same direction as the
moving particle: W =
approximate with For a small subsection of the curve , the force is approximately constant and we can . ( points in the direction of movement and is of length one) Thus we have the work for a subsection of the curve C: NOTATION ALERT!!! ((the above represent the line integrals with respect to x, y, and z)) The definition for work is not very useful since it is parameterized with arc length. But the equivalent definition is useable since it does not rely on arc length parameterization. Example:
Prove one of the Basic Laws of Mechanics, the Work-Kinetic Energy Principle:
The work done equals the change in kinetic energy Use Newton’s Second Law: Now use the fact that The next example will show you how to deal with line integrals of the form: Example:
Let and the path of integration C: Find the Work done by the vector field. We must find . and substitute them into the...
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