What is Integration? 3 Ways to Interpret Integrals
Summary
TLDRこの動画スクリプトでは、積分を誤った理解を持っているという主張を行います。学生たちは積分を曲線下の面積、反導関数、または微量を足し合わせることの3つの意味の1つと捉えがちですが、実際には微量を足し合わせることの概念が実世界の問題解決に役立ちます。動画では、弓の引き力や太陽光発電、バスケットボールの走行距離など、実例を通じて積分を理解する方法を説明し、積分における誤った理解を修正し、実用的なスキルを身につけることができるよう導きます。
Takeaways
- 📚 積分を誤解しているという主張の根拠は、微積分から卒業した学生たちの認識についての研究に基づいています。
- 🧠 学生たちは積分を3つの意味として概念化しています:曲線下の面積を見つけること、関数の反導関数を求めること、そして微量を加算すること。
- 🔢 研究によると、曲線下の面積や反導関数だけを考え的学生は、積分が必要とする実際の問題を解決する際に苦労します。
- 🌟 成功した学生は、微量を加算するという積分の概念を理解しており、新しい状況で積分が使える場合を認識し、問題解決に役立てています。
- 📈 積分を正しく理解するためには、微量を加算するという考え方が必要です。これは、初年度の微積分で、数量を掛け算によって計算することを意味します。
- 🏹 弓の例では、弓を引くにつれてエネルギーが増加し、全体的なエネルギーを計算する方法が説明されています。
- 🌞 太陽光発電パネルの例では、パネルが太陽を追いかけるトラッカーに設置されている場合と、屋上に平らに設置されている場合のエネルギー獲得の違いが考慮されています。
- 🏀 バスケットボールの例では、快攻時にサイドに走るよりも直接走った方が有利であるかどうかを検討しています。
- 📊 積分の3つの問題状況は、調整して単純な掛け算問題に変形でき、それらを用いて各問題を解決する戦略を形成できます。
- 🔧 積分においては、短い時間間隔または距離間隔をたくさんの小さな区間に分割し、各区間に関連する数量を計算し、それらを足して総額を求めます。
- 📝 積分を理解するためのこの動画は、積分を面積として考えることの代わりに、微量を加算するというより強力な方法を提供しています。
Q & A
学生が積分をどのように誤解しているのか説明してください。
-多くの学生は、積分を曲線下の面積を見つけること、または関数の反微分を求ることと考えます。しかし、実際には、積分は微量を加算することを表しており、これが実世界の問題解決に役立ちます。
積分を正しく理解するために、どのような3つの問題を例に挙げられましたか?
-1) 弓を引くときのエネルギー計算、2) ソーラーパネルの設置方法によるエネルギーの変化、3) バスケットボールの快攻における走行距離の比較です。
弓の例で示されたデータは何を表しているのですか?
-そのデータは、弓を引くにつれ1.5インチ(約0.3メートル)ごとに必要な力(lbs)を表しており、合計18インチの引っ張りの中で12つの区間で力の値を測定しています。
弓のエネルギーを計算する際、どのような手順を取りましたか?
-まず、弓を引くことによってエネルギーを計算する区間を小分けにし、それぞれの区間に適用される力を平均値で見積もります。次に、それぞれの区間に適用される力を距離で掛けてエネルギーを計算し、すべての区間のエネルギーを合計します。
実世界の問題において、積分を正しく認識し、適切な積分を設定するためには、どのような考え方が重要ですか?
-積分を微量の加算として考えることが重要です。これにより、新しい状況で積分が使用可能であるかどうかを認識し、適切な積分を設定することができます。
積分記号が示す処理は何ですか?
-積分記号は、状況を非常に小さく均等な区間に分割し(微分dxやdtと表される)、各区間の関数の値をその区間の幅と掛けて計算し、それらをすべて足し合わせることを意味しています。積分記号は「summa」を表す「s」の伸ばしで、レイニッツによって選ばれました。
教科書ではなぜ面積の概念で積分を説明するのですか?
-面積の概念で積分を説明することで、抽象的な積分を理解しやすくなります。例えば、弓の設計者は弓の力の曲線下の面積を用いて理想的な弓の特性を考慮します。
弓の例で、実際のデータを使って得られたエネルギーの見積もりと、モデルを使って得られた見積もりはどの程度一致していますか?
-実際のデータを使った見積もりでは10.425 foot-lbsのエネルギーが必要であったのに対し、モデルを使って計算した見積もりでは10.3275 foot-lbsが必要であるため、非常に近い見積もりとなっています。
このビデオスクリプトにおいて、積分を正しく理解するための最も重要なポイントは何ですか?
-積分を正しく理解するための最も重要なポイントは、積分を微量を加算することを表すという考え方を持つことです。これにより、実世界の問題を認識し、適切な積分を設定することができます。
このスクリプトで提案された積分の考え方はどのようにして実世界の問題に適用されるのですか?
-このスクリプトで提案された積分の考え方は、実世界の問題を小さく分割し、それぞれの小さく chop up た部分で簡単に計算可能なものにすることで適用されます。そして、これらの計算結果を加算して、全体の解決策を得ることができます。
このビデオスクリプトでは、どのような方法で弓の力を測定しましたか?
-弓を引くことで必要な力を測定し、1.5インチごとに力の値を平均値で見積もりました。そして、距離の測定を英尺に変換して、足くのlbをエネルギー単位として使用しました。
このビデオスクリプトでは、積分を計算する際に反微分を求める方法と、数値積分計算器を使用する方法のどちらがより正確であると言えますか?
-どちらの方法も正確な答えを得るために是用いられますが、このビデオスクリプトでは、実際のデータに基づく近似解答を得るために、数値積分計算器を使用する方法が提案されています。
Outlines
📚 誤った積分概念の修正
この段落では、多くの学生が誤った積分概念を学んでいるという主張を説明しています。積分を曲線下の面積、反導関数、または微量を足し合わせる行為とのみ捉えがちですが、実際には実世界の問題解決において、微量を足し合わせるという考え方が重要であることが指摘されています。この視点で積分を理解し、初めて実用的な問題解決が可能になると述べています。また、この方法が CALC I での例えを通じて説明されており、その応用方法が詳細に解説されています。
🔍 積分の核心プロセスとその近似解
この段落では、積分の核心プロセスである微小な区間での近似とその近似解の計算方法について説明されています。実際のデータに基づく積分問題の場合、完全な解を得ることができないため、近似値を使用することが重要です。このプロセスを通じて、弓の引き力とエネルギーの例を用いて、積分を近似する方法が詳細に解説されています。また、多項式曲線フィットと積分符号の意味についても触れられています。
🏹 弓の例による積分の視覚化と面積の考え方
最後の段落では、弓の例を通じて積分を視覚化し、面積に基づく積分の考え方について説明されています。弓の設計における力の曲線下の面積を用いた理論と、実世界の問題解決における面積考え方の利点と欠点を比較しています。また、積分問題を認識し、積分式を設定するための「切る、掛ける、足す」というプロセスの重要性が強調されています。
Mindmap
Keywords
💡integration
💡antiderivative
💡real-world problems
💡quantities
💡multiplication
💡intervals
💡force
💡energy
💡solar panels
💡basketball
💡approximation
Highlights
Students often misunderstand integration, focusing on finding areas under curves or antiderivatives, which can hinder their ability to solve real-world problems.
Integration can also be conceptualized as adding up tiny-bits of a quantity, a perspective that is beneficial for tackling real-world applications.
The traditional calculus education may not equip students with the tools to recognize when integration can be applied in novel situations.
Students who view integration as adding up quantities are more likely to succeed in solving complex problems involving integration.
The video aims to help viewers understand integration as a powerful tool for problem-solving by focusing on the concept of adding up small quantities.
Three problem situations are presented to illustrate the concept of integration: energy stored in a bow, energy output from solar panels, and distances run in basketball.
These problem situations are simplified to multiplication problems, demonstrating how integration can be approached with basic math concepts.
The弓 (bow) example is used to demonstrate how to break down a complex situation into small intervals for easier calculation.
Data from a toy compound bow is used to calculate the force required to pull the string back at different intervals.
The process of integration involves chopping up a situation into small intervals, calculating the quantity within each interval by multiplication, and summing these quantities for the total value.
The concept of integration is deeply rooted in multiplication and summation, which is reflected in the structure of integral notation.
Integration can be approximated using numerical methods, such as fitting a model to data points and calculating the total energy based on that model.
The fundamental theorem of calculus can be used for integration when a function is continuous and an antiderivative can be found.
While thinking of integration as areas can be helpful in some contexts, it is not as effective for recognizing and setting up integration problems in real-world scenarios.
The video encourages viewers to adopt a new perspective on integration that can help them identify and solve problems in their everyday lives.
The bow example is used to show how the process of integration can be quantitatively understood by breaking up the distance into small pulls and calculating the energy for each.
Integration as a concept is not just about areas but is a versatile tool for problem-solving that involves the process of chop, multiply, and add.
Transcripts
You learned integration wrong. That's quite a claim, so let me clarify what we mean by it.
Our claim is based on studies of former calculus students that analyzed what students think
integration means. Students tend to conceptualize integration to have one of three meanings:
Integration is finding an area under a curve, integration is finding an antiderivative of
a function, and integration is adding up tiny-bits of a quantity.
Now we can think about integration as all of these, but research shows that students
who only think of integration as finding areas under curves or finding antiderivatives really
struggle to solve real-world problems where integration is required. But which students
did succeed in solving these same problems? The students that could think of integration
as adding up tiny-bits of a quantity. So why do we claim you learned integration
wrong? Well because most students leave calculus with one of the first two conceptions, areas
under curves or anti-derivatives. And, to be fair, these are the conceptions mostly emphasized
in calculus classes. But this leaves the vast majority of students without the tools to
recognize when integration could be used in a novel situation, and even if they do recognize
the use of integration, they still get stumped trying to set up the appropriate functional
or numerical integral to solve the problem. So this is what I mean by "you learned integration
wrong." You probably don't think about integration in a way that actually makes it useful for
real world problem solving. This video will help you understand integration
in this more powerful way, adding up tiny bits of a quantity. For first year calculus,
the quantities are all calculated with multiplication, so we use these examples in our video.
This video is a bit longer than our other videos but we hope you find it worth it.
Here are three different problem situations to illustrate integration:
1) When I pull back my bow to shoot an arrow, it gets harder and harder to pull, so how much
total energy do I put into the bow if I pull it all the way back?
2) How much more energy could I get from solar panels if I put them on solar trackers instead
of laying them flat on a roof? 3) In basketball, my coach would tell me to
run wide on the fast break, but wouldn't it be better if I ran straight down the court
instead? Doesn't going wide give my defender more time to beat me down the court and set-up defenses?
Since I play near the basket, I need to know how much farther I run going from
basket to basket out wide, versus in a straight line.
These 3 problem situations are almost 6th grade math problems, what I mean by that,
is that we could adjust each of these problems slightly and end up just doing multiplication.
In reality the force to pull a bow back increases the farther back I pull. But if the force
was constant, like lifting a book from the floor to a table, then calculating the energy
would just be a multiplication problem. The force times the distance.
Our solar panel is stagnant. But if the solar panel moved to always face the sun, assuming
a sunny day, then the power generated is almost constant, so I can multiply the power rating
of the solar panel by the amount of time it's in the sun.
If I always ran at the same angle relative to my basketball defender, rather than along
this arc like I would in reality, then it's easy to find out how much further I run compared
to my defender. I just multiply his distance by a certain ratio
We can build on these simple cases to develop a strategy for solving each of these problems.
Chop up each situation into a lot of small intervals, where the problems are quite close
to the multiplication problems described above, do the multiplication in each of these small
intervals to calculate the quantity we're interested in (in these cases energy or distance),
then add up the values from each small piece to get a total amount.
The idea is that for a short interval in time, or distance, the difference between the function
value at the beginning of the interval and the end of the interval is so small the effect
on the final answer is negligible so we can treat them as constant and just use multiplication
The force of pulling the bow doesn't change much from one millimeter to the next. The power
generated from the solar panel doesn't change much from one minute to another. The ratio
of my distance to my defenders distance is about the same from any point and a point
one centimeter further down the court. So for each situation, we can break it up
into a lot of small intervals, do the multiplication to calculate the amount in each interval,
and then add up the amounts from each interval.
Let's dive deeper in my bow example. Here is data from a toy compound bow I bought at
a garage sale to use in demonstrations in my college math classes.
I calculated the force, in lbs, it took to hold the string back every 1.5 inches of the
pull, which is a total of 18 inches. I have 12 intervals, so 13 positions where I collected
force values. I am going to use the average value of each
end point of the interval to estimate the average lbs of force to pull the string on
that interval. I will convert the distance measurements to feet, so I end up with foot-lbs
as the energy unit. Within each 1.5 inch interval, or 1/8th of
a foot, I can calculate the energy to pull back the bow by multiplying the average force
times the distance. Then I can calculate the total amount of energy by adding up the energy
expended in each of the intervals. I got a value of 10.425 foot-lbs. So enough to hurt
someone... well not that bad. More like dropping a 1
lb weight from a height of 10 ft on someone's foot.
This process we just went through is integration at. its. heart. We first: Chopped up a situation
where there isn't a constant relationship between quantities, so that within each interval
it is approximately constant. Second: we calculated the quantity we are trying to find within
each interval by multiplying, and 3) added up the quantity in each piece for the total value.
You may have noticed, we didn't write down an integral sign, though
we could have. We didn't find an anti-derivative. We actually don't have a function to even
find the anti-derivative of. But that's the case in a lot of real world integration problems
when we are working with data. Now you may be thinking that my answer is
not exact. Well, I would say you are correct, but in this case, no one could be exact.
We could certainly have used a lot more measurements on smaller intervals to be more accurate with our data,
but at some point there will be more error in trying to pull the string with the scale
and hold it exactly at the right distance, or in the truncated digits of the scale, than
in the mathematical calculations. We could use some assumptions to get around
some of the physical limitations of taking measurements, but the assumptions don't mean
that we are still going to get the exact answer. For example, we could fit a model to the values
I measured earlier. Here is a 4th degree polynomial fit to our points, where the distance the
bow is pulled in feet is on the x-axis and the force in lbs on the y-axis. It seems to
fit quite well (with an r-squared of over .98 for those interested).
So we could calculate the total energy assuming that the force to pull the bow at any point
is the value given by the function. This gets us around the measurement issues, but of course,
our model won't fit the real-world phenomenon, exactly, so we are still just approximating,
but for different reasons than before. If you have taken calculus before, you might
be more comfortable setting up and calculating the energy based on this model with a function.
But the thinking to set up the integral is still the same as before.
But, first, let's connect this idea of multiplication and summation to the integral notation you
may be familiar with. The structure of integral notation actually
suggests the multiplication and summation processes central to integration. We chop
up the situation into arbitrarily small intervals of equal width called the differential, often
labeled dx or dt. Then we multiply the function value on the interval with the width of the
interval, and finally add it all up. The integral sign is actually an elongated
s, chosen by Leibniz to symbolize "summa" which means total or sum.
It's interesting how once we know what each piece of the integration notation represents, how
easy it is to see that the core concept of integration is this process of multiplication
and summation. And yet this conception's not emphasized and often overlooked in so
many calculus classes. Let's get back to the bow example. So in order to quantitatively understand this
situation we'll chop up the distance into a bunch of small pulls. How much energy is
any one of the pulls? It is the force you feel times the distance you pull.
Since we have a function that can tell us the force at any value, we can then pick extremely
small intervals, infinitesimally small intervals, so the differential, dx, is the length of
each interval. What are all of the values for which we accumulated energy? Well, our
pull goes from zero feet to 1.5 feet, so that is the interval we are integrating over. And
finally, adding up all of those products that we could calculate over that interval and
we get our final answer. Since we have a function that connects a force
at any point on the pull, we could use a spreadsheet to do the calculations. Or use any one of the
many online numerical integration calculators, where people have programmed computers to
do it for us. This is also one of the special cases where
we could use the fundamental theorem of calculus. Since our function is continuous on our interval
of integration, and we can find an antiderivative, then we can use an antiderivative to make
the calculation. Why does this antidifferentiation technique
work? That's beyond the scope of this video. But perhaps we'll address it in another video
in the future. Comment below if you'd like to learn more on why antidifferentiation works!
Whether using the Fundamental Theorem of Calculus or using an integration calculator, with this model we get an estimated
force of 10.3275, quite close to our first estimate with 12 sub-intervals.
Now let's talk about integrals as areas. It is true that each of the 3 integration situations
can be connected to a problem about areas. The problem is that thinking about areas doesn't
help to recognize any of those problems as an integration problem in the first place.
But thinking about integration as the chop, multiply, and Add process, helps with both recognizing an integration
problem and setting up the integrand. Perhaps the problem that is closest to being
made sense of as an area in an authentic way, true to the science of the situation, is the
one we did with the bow, at least once we had a function to model the force.
There are times when thinking about integrals as areas is helpful. Engineers that design
compound bows use the area under the force curve of the bow to reason about the properties
of an ideal bow. And since any integration problem can be thought
of as an area problem, then areas can help us visually understand properties of integrals,
and strategies for approximating integrals. I think this is one reason why textbooks emphasize
the area conception so much, because it is helpful in making sense of abstract integrals.
Thinking about integration as areas unfortunately is not very helpful at the front end of real-world
problems where we need to recognize the problem as an integration problem and set up the correct integral.
So we hope now you can start to see integration
in a new way, if you haven't been thinking of it this way before. If you can develop
this way of thinking, you might find, like I have, that you start to notice integration
problems all around you.
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