Difference between revisions of "Calc3.8"
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=The Line Integral= | =The Line Integral= | ||
| − | + | ==Definition== | |
We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space. But lines, and curves, can be contained in the plane, and surfaces can be contained in space. Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains? | We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space. But lines, and curves, can be contained in the plane, and surfaces can be contained in space. Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains? | ||
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<math>\int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k </math> | <math>\int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k </math> | ||
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| + | ==Computation== | ||
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| + | As before, we're not going to be using the Riemann sum definition to compute line integrals. Let <math>x=x(t),y=y(t) </math> be a parametric description of <math>C </math>. This allows us to compute the length element, ''dL'': | ||
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| + | <math>dL = \sqrt{dx^2+dy^2} </math> | ||
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| + | But we don't have any formulas for ''dx'' or ''dy''! The closest we've got are formulas for <math>dx/dt, dy/dt</math>; we'll have to plug these in and see if we can patch things up: | ||
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| + | <math>= \sqrt{ \left({ \frac{dx}{dt} }\right)^2 + \left({ \frac{dy}{dt} }\right)^2 } \sqrt{dt^2} </math> | ||
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| + | Now we can just plug in our formula for ''dL'' into the integral to get an easy-to-use formula for a line integral: | ||
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| + | <math>\int_C f(x,y)dL = \int_{t_0}^{t_1} f(x,y)\sqrt{ \left({ \frac{dx}{dt} }\right)^2 + \left({ \frac{dy}{dt} }\right)^2 } dt </math> | ||
=Green's Theorem in the Plane= | =Green's Theorem in the Plane= | ||
Revision as of 08:07, January 13, 2010
Generalized Integration
The Line Integral
Definition
We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space. But lines, and curves, can be contained in the plane, and surfaces can be contained in space. Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains?
Well, because it is simpler, and useful to have under our belts before we expand our repetoire. But there is no reason we can't integrate a function over a region in its domain which is a lower dimension than the domain itself.
We'll start with the line integral. Technically, it should be called the curve integral, but let's let that slide for now and get to the definition. We're trying to integrate over a curve, say, C. At first, we'll let this curve lie in the xy plane - it's not difficult to generalize to three dimensions. So we proceed as before: we divide this region into n pieces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_k, 1\leq k \leq n} , and we pick a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_k,y_k)\in C_k} for each piece. Letting the length of be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_k} . The definition follows as expected:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k }
Computation
As before, we're not going to be using the Riemann sum definition to compute line integrals. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x(t),y=y(t) } be a parametric description of . This allows us to compute the length element, dL:
But we don't have any formulas for dx or dy! The closest we've got are formulas for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx/dt, dy/dt} ; we'll have to plug these in and see if we can patch things up:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sqrt{ \left({ \frac{dx}{dt} }\right)^2 + \left({ \frac{dy}{dt} }\right)^2 } \sqrt{dt^2} }
Now we can just plug in our formula for dL into the integral to get an easy-to-use formula for a line integral:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_C f(x,y)dL = \int_{t_0}^{t_1} f(x,y)\sqrt{ \left({ \frac{dx}{dt} }\right)^2 + \left({ \frac{dy}{dt} }\right)^2 } dt }
Green's Theorem in the Plane
The Surface Integral
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