visualizing the vector averageexecute this worksheet first to use the Explore slider at the endIn Calc 2 we learn that the average value of a function is its integral divided by the length of the interval over which the averaging takes place: 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where 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 is the width of the subintervals of the Riemann sum approximation to the definite integral used in the limit expressions while 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 is the midpoint of the i th subinterval making this the midpoint approximation.
The final equality shows that the average is just the numerical average of the n sampled midpoint values of the function.The same interpretation gives meaning to the integral of a vector function. It averages out the position vector of the tip of the vector valued function as it sweeps out a spacecurve in space. The vector average: 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is just the vector of averages of the vector's components, but its interpretation is geometrical.Here is an example.creating a procedure to visualize a discrete Riemann 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We just make a procedure of the integral divided by the length of the integration interval.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 we visualize the endpoint position vectors (green) and the average position vector (blue) for the spacecurve which traces out the vector valued function.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Finally we make an approximate average using only n sampled points, the midpoints of the 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The black arrow is the approximate average over 1 to 5 sampled points, to compare with the exact blue vector average. It is just the midpoint value of the position vector when there is only 1 subinterval. It quickly approaches the average.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
SSJtRzYi#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plots:-display(ShowVectorAverage(F,0,Pi),plots:-arrow(VectorAverageDiscrete(F,0,Pi,m),shape = plots:-arrow,color = black)), ':-Plot0', [':-m'], ':-sliders'=["Sliderm"], ':-labels'=["Labelm"],':-isplot'=true, ':-handlers'=[m=handlertab[m]]);
#
# End of autogenerated code.
#
It appears that only a few divisions (move the slider to the right) make the black approximate average vector overlap the exact average (blue) for many curves. If you re-execute this with a new curve and endpoints, you can play around with it.We can do better. We can visualize the discrete set of sampled vectors used to average the position vector.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJHNlcUYnRi9GMi1GNjYkLUYjNi0tSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlgtRiw2JVEmYXJyb3dGJ0YvRjItRjY2JC1GIzYuLUYsNiVRJXN1YnNGJ0YvRjItRjY2JC1GIzY0LUYsNiVRImFGJ0YvRjItRkI2LVEiPUYnRkVGR0ZKRkxGTkZQRlJGVC9GV1EsMC4yNzc3Nzc4ZW1GJy9GWkZqby1JI21uR0YkNiRRIjBGJ0ZFLUZCNi1RIixGJ0ZFRkcvRktGMUZMRk5GUEZSRlRGVi9GWlEsMC4zMzMzMzMzZW1GJ0ZBLUYsNiVRImJGJ0YvRjJGZm8tRiw2JVEnJiM5NjA7RidGL0YyLUZCNi5GYnBGL0YyRkdGY3BGTEZORlBGUkZURlZGZHAtRiw2JVEibkYnRi9GMkZmby1GXXA2JFEiM0YnRkVGYHBGQS1GLDYlUSJGRidGL0YyLUY2NiQtRiM2K0Zjby1GQjYtUSIrRidGRUZHRkpGTEZORlBGUkZUL0ZXUSwwLjIyMjIyMjJlbUYnL0ZaRl9yLUY2NiQtRiM2Ji1GLDYlUSJpRidGL0YyLUZCNi1RKCZtaW51cztGJ0ZFRkdGSkZMRk5GUEZSRlRGXnJGYHItSSZtZnJhY0dGJDYoLUZdcDYkUSIxRidGRS1GIzYkLUZdcDYkUSIyRidGRUZFLyUubGluZXRoaWNrbmVzc0dGYHMvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGanMvJSliZXZlbGxlZEdGSUZFRkVGQS1GQjYtUScmc2RvdDtGJ0ZFRkdGSkZMRk5GUEZSRlRGVkZZLUZcczYoLUY2NiQtRiM2JkZmcEZockZjb0ZFRkUtRiM2JEZecUZFRmZzRmhzRlt0Rl10RkEvJStleGVjdXRhYmxlR0ZJRkVGRUZqdEZFRkVGYHAtRiw2JVEmc2hhcGVGJ0YvRjJGZm9GZW5GYHAtRiw2JVEmY29sb3JGJ0YvRjJGZm8tRiw2JVEkcmVkRidGL0YyRmp0RkVGRUZgcEZlckZmb0Zecy1GQjYtUSMuLkYnRkVGR0ZKRkxGTkZQRlJGVEZeckZZLUZdcDYkUSI0RidGRUZqdEZFRkVGanRGRUZFRmp0RkU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVE6VmVjdG9yQXZlcmFnZURpc2NyZXRlUGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSomY29sb25lcTtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGUy1GLDYvUSVwcm9jRicvJSVib2xkR0ZML0YwUSVib2xkRicvJStmb250d2VpZ2h0R0ZlbkYyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JC1GIzYqLUZHNiVRIkZGJ0ZKRk0tRiw2LVEiLEYnRi9GMi9GNkZMRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUZHNiVRImFGJ0ZKRk1GYG8tRkc2JVEiYkYnRkpGTUZgby1GRzYlUSJuRidGSkZNRi9GL0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRkMvJSZkZXB0aEdGZHAvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRmBwNiZGYnBGZXBGZ3AvRmpwUSVhdXRvRictRkc2JVEoZGlzcGxheUYnRkpGTS1GaW42JC1GIzYlLUZHNiVRJHNlcUYnRkpGTS1GaW42JC1GIzYsRistRkc2JVEmYXJyb3dGJ0ZKRk0tRmluNiQtRiM2LkYrRl1vLUZpbjYkLUYjNipGZm8tRiw2LVEiK0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZdcy1GaW42JC1GIzYmLUZHNiVRImlGJ0ZKRk0tRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GP0Zcc0Zecy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGLy1GIzYkLUZddDYkUSIyRidGL0YvLyUubGluZXRoaWNrbmVzc0dGX3QvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaXQvJSliZXZlbGxlZEdGNEYvRi9GKy1GLDYtUScmc2RvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUZqczYoLUZpbjYkLUYjNiZGaW9GZnNGZm9GL0YvLUYjNiRGXHBGL0ZldEZndEZqdEZcdUYrRi9GL0Zgby1GRzYlUSZzaGFwZUYnRkpGTS1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/RlJGVEZeckZgby1GRzYlUSZjb2xvckYnRkpGTUZcdi1GRzYlUSRyZWRGJ0ZKRk1GL0YvRmBvRmNzRlx2Rlx0LUYsNi1RIy4uRidGL0YyRjVGN0Y5RjtGPUY/RlxzRkRGXHBGL0YvRi9GL0YrRistRiw2L1EkZW5kRidGWEZaRmZuRjJGNUY3RjlGO0Y9Rj9GQUZERitGVS1GLDYtUSI6RidGL0YyRjVGN0Y5RjtGPUY/RlJGVC8lK2V4ZWN1dGFibGVHRjRGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVE6VmVjdG9yQXZlcmFnZURpc2NyZXRlUGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYrLUZHNiVRIkZGJ0ZKRk0tRiw2LVEiLEYnRi9GMi9GNkZMRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMEYnRi9GVy1GRzYlUScmIzk2MDtGJ0ZKRk0tRiw2LkZZRkpGTUYyRlpGN0Y5RjtGPUY/RkFGZW4tRmhuNiVRIjRGJ0ZKRk0vJStleGVjdXRhYmxlR0Y0Ri9GL0Zjb0YvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYjNiUtSSNtaUdGJDYlUShFeHBsb3JlRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjItRkk2JVEoZGlzcGxheUYnRkxGTy1GUjYkLUYjNjQtRkk2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGQy8lJmRlcHRoR0Zfby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GW282JkZdb0Zgb0Ziby9GZW9RJWF1dG9GJy1GSTYlUTJTaG93VmVjdG9yQXZlcmFnZUYnRkxGTy1GUjYkLUYjNigtRkk2JVEiRkYnRkxGTy1GLDYtUSIsRidGL0YyL0Y2Rk5GN0Y5RjtGPUY/RkEvRkVRLDAuMzMzMzMzM2VtRictSSNtbkdGJDYkUSIwRidGL0ZlcC1GSTYlUSUmcGk7RicvRk1GNEYvRi9GL0ZlcC1GSTYlUTpWZWN0b3JBdmVyYWdlRGlzY3JldGVQbG90RidGTEZPLUZSNiQtRiM2KkZicEZlcEZbcUZlcC1GSTYlRmFxRkxGTy1GLDYuRmdwRkxGT0YyRmhwRjdGOUY7Rj1GP0ZBRmlwLUZJNiVRIm1GJ0ZMRk9GL0YvRmVwRmpuRmdvLUZJNiVRJmFycm93RidGTEZPLUZSNiQtRiM2LS1GSTYlUTZWZWN0b3JBdmVyYWdlRGlzY3JldGVGJ0ZMRk8tRlI2JC1GIzYrRmJwRmVwRltxRmVwRl9xRmVwRitGXnJGL0YvRmVwLUZJNiVRJnNoYXBlRidGTEZPLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGZnNGYXJGZXAtRkk2JVEmY29sb3JGJ0ZMRk9GYnMtRkk2JVEmYmxhY2tGJ0ZMRk9GL0YvRmduRmpuRmdvRmduRi9GL0ZlcEZqbkZnb0ZeckZicy1GXHE2JFEiMUYnRi8tRiw2LVEjLi5GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRidGRC1GXHE2JFEiNUYnRi9GK0ZlcC1GSTYlUSpwbGFjZW1lbnRGJ0ZMRk9GYnMtRkk2JVEmcmlnaHRGJ0ZMRk9GL0YvRi9GZ24vJStleGVjdXRhYmxlR0Y0Ri8=
SSJtRzYi#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plots:-display(ShowVectorAverage(F,0,Pi),VectorAverageDiscretePlot(F,0,Pi,m),plots:-arrow(VectorAverageDiscrete(F,0,Pi,m),shape = plots:-arrow,color = black)), ':-Plot1', [':-m'], ':-sliders'=["Sliderm1"], ':-labels'=["Labelm1"],':-isplot'=true, ':-handlers'=[m=handlertab[m]]);
#
# End of autogenerated code.
#
The endpoints of the semicircle are the green arrows. The red arrows are the midpoint values in the n subintervals. For example when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8it shows the midpoint of the semicircle. The approximate average very quickly approaches the blue limit.To visualize the original integral (not the average value), one need only multiply the average by the parameter interval width.TTdSMApJQVJUQUJMRV9TQVZFLzM2ODkzNDkwNDQ2MjgyNTc4MjY4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiEsJCokJSNQaUchIiIiIiNGLkYl