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#Sumation Reel by @mathswithmuza - A sum is about adding up individual pieces that stay separate. You take a finite or countable set of values and combine them one by one, like adding t
963.2K
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@mathswithmuza
A sum is about adding up individual pieces that stay separate. You take a finite or countable set of values and combine them one by one, like adding the heights of a bunch of thin rectangles. Each term is distinct, and nothing happens in between those points. Sums naturally show up when you’re working with discrete data, step-by-step processes, or situations where the pieces are clearly separated, such as counting objects, adding probabilities in a table, or approximating areas using rectangles. The result depends on how many pieces you include and how wide or narrow each piece is. An integral, on the other hand, treats the quantity as continuously changing. Instead of adding individual chunks, integration blends infinitely many tiny contributions into a single total. You can think of it as what happens when a sum is pushed to the extreme: the pieces become infinitely thin and the approximation turns exact. This makes integration the natural tool for finding areas under curves, total distance from a changing velocity, or accumulated quantities in physics and economics. While a sum feels like stacking blocks, an integral feels like filling a shape with no gaps at all. Like this video and follow @mathswithmuza for more! #math #maths #integration #foryou #3d
#Sumation Reel by @learntechbyadi - A sum is about adding up individual pieces that stay separate. You take a finite or countable set of values and combine them one by one, like adding t
2.1K
LE
@learntechbyadi
A sum is about adding up individual pieces that stay separate. You take a finite or countable set of values and combine them one by one, like adding the heights of a bunch of thin rectangles. Each term is distinct, and nothing happens in between those points. Sums naturally show up when you’re working with discrete data, step-by-step processes, or situations where the pieces are clearly separated, such as counting objects, adding probabilities in a table, or approximating areas using rectangles. The result depends on how many pieces you include and how wide or narrow each piece is. An integral, on the other hand, treats the quantity as continuously changing. Instead of adding individual chunks, integration blends infinitely many tiny contributions into a single total. You can think of it as what happens when a sum is pushed to the extreme: the pieces become infinitely thin and the approximation turns exact. This makes integration the natural tool for finding areas under curves, total distance from a changing velocity, or accumulated quantities in physics and economics. While a sum feels like stacking blocks, an integral feels like filling a shape with no gaps at all. Like this video and follow @learntechbyadi for more! #math #maths #integration #foryou #3d
#Sumation Reel by @dybydx.ai - A sum is about adding up individual pieces that stay separate. You take a finite or countable set of values and combine them one by one, like adding t
10.0K
DY
@dybydx.ai
A sum is about adding up individual pieces that stay separate. You take a finite or countable set of values and combine them one by one, like adding the heights of a bunch of thin rectangles. Each term is distinct, and nothing happens in between those points. Sums naturally show up when you’re working with discrete data, step-by-step processes, or situations where the pieces are clearly separated, such as counting objects, adding probabilities in a table, or approximating areas using rectangles. The result depends on how many pieces you include and how wide or narrow each piece is. An integral, on the other hand, treats the quantity as continuously changing. Instead of adding individual chunks, integration blends infinitely many tiny contributions into a single total. You can think of it as what happens when a sum is pushed to the extreme: the pieces become infinitely thin and the approximation turns exact. This makes integration the natural tool for finding areas under curves, total distance from a changing velocity, or accumulated quantities in physics and economics. While a sum feels like stacking blocks, an integral feels like filling a shape with no gaps at all. #math #maths #integration #foryou #3d
#Sumation Reel by @mathswithmuza - Integration is one of the central ideas in calculus, and at its core it is about accumulation. When we talk about integration in the context of area,
77.6K
MA
@mathswithmuza
Integration is one of the central ideas in calculus, and at its core it is about accumulation. When we talk about integration in the context of area, we are usually thinking about the area under a curve on a graph. Imagine a function drawn on coordinate axes. If you want to know the area between that curve and the horizontal axis over some interval, you can approximate it by slicing the region into many thin rectangles. Each rectangle has a small width and a height determined by the function. Adding up the areas of all these rectangles gives an approximation of the total area. Integration is what happens when those rectangles become infinitely thin, so the approximation becomes exact. This idea connects directly to the definite integral. The definite integral over an interval represents the net area between the curve and the axis on that interval. If the function stays above the axis, the integral equals the usual geometric area. If the function dips below, the integral subtracts that part, giving what is called signed area. In this way, integration provides a precise mathematical way to measure area under curves, even when the shapes are irregular and cannot be handled by simple geometry. Like this video and follow @mathswithmuza for more! #math #calculus #visualization #animation #foryou
#Sumation Reel by @bytebrain.tv - Estimating area and volume is one of the foundational problems that led to the development of calculus. The key idea begins with summation: breaking a
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@bytebrain.tv
Estimating area and volume is one of the foundational problems that led to the development of calculus. The key idea begins with summation: breaking a complex shape into many smaller, simpler pieces and adding them together. Using summation, the region under a curve or surface can be approximated by dividing the domain into a finite number of subintervals. Over each interval, a rectangular (or box-shaped) approximation is formed, with its height determined by the function value at a chosen point. Adding the areas or volumes of these pieces produces a Riemann sum, which provides an estimate of the total region. However, because curved boundaries are being approximated with flat shapes, the result contains error. The approximation improves as more subdivisions are used, but it remains an estimate as long as the pieces have nonzero width. Integration formalizes this process by taking summation to its ultimate limit. Instead of using a fixed number of rectangles, the width of each subdivision is allowed to shrink toward zero. As the number of pieces approaches infinity, the summation becomes an integral, and the approximation becomes exact. Integration therefore represents the precise accumulation of infinitely many infinitesimal contributions, capturing continuous change in a way that finite summation cannot. In higher dimensions, the same principle applies: volumes under surfaces can be estimated using 3D sums of small boxes, and calculated exactly through double or triple integrals. Integration is essentially the perfected form of summation, providing exact solutions in physics, engineering, and geometry. #Calculus #Integration #Summation #RiemannSum #MultivariableCalculus #3DMath #VolumeUnderSurface #AreaEstimation #STEMEducation #EngineeringMath #MathematicalModeling #HigherMath #Physics #ContinuousSystems #MathConcepts
#Sumation Reel by @phyxon_17 - Integration vs Summation. Continuous vs Discrete. Same sine curve - different mathematics. What's the difference between: ∫ sin(x) dx and Σ sin(x) Th
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@phyxon_17
Integration vs Summation. Continuous vs Discrete. Same sine curve — different mathematics. What’s the difference between: ∫ sin(x) dx and Σ sin(x) They may look similar, but they are fundamentally different. 1️⃣ Integration of sin(x) ∫ sin(x) dx = −cos(x) + C Integration gives the continuous accumulated area under the sine curve. Graphically: You are measuring smooth area under a smooth wave. 2️⃣ Summation of sin(x) Σ sin(xₖ) Summation adds discrete sampled values of the sine function. Graphically: You are stacking vertical bars at chosen points. Key Insight 💡 If you take more and more sample points, the summation starts to approximate the integral. This is the foundation of: • Numerical methods • Riemann sums • Engineering simulations • Signal processing • Physics modelling Continuous mathematics meets discrete computation. That’s how calculus powers modern technology. ENGAGEMENT 👇 If you increase the number of sample points, does the summation get closer to the integral? Yes or No? Comment below. #Calculus #RiemannSum #EngineeringMath #STEMEurope 🇪🇺 #AppliedMathematics SignalProcessing UniversityMath PhysicsStudents FutureEngineers PhyxonAcademy DM “EUROPE” for structured Maths & Physics coaching Follow @phyxonacademy for serious STEM clarity Limited slots. Dedicated students only.
#Sumation Reel by @degamma_maths - Γ(z) v/s ζ(z) 🫧 #integral #integration #calculus #stem #maths
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DE
@degamma_maths
Γ(z) v/s ζ(z) 🫧 #integral #integration #calculus #stem #maths
#Sumation Reel by @equationacademy - ➡️Visualizing Differentiation of Sin^50x ➡️ Follow @equationacademy for more #mathematics #differentiation #calculus #fblifestyle #science
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EQ
@equationacademy
➡️Visualizing Differentiation of Sin^50x ➡️ Follow @equationacademy for more #mathematics #differentiation #calculus #fblifestyle #science
#Sumation Reel by @degamma_maths - Satisfying 🫧 #integral #integration #maths #stem #calculus
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@degamma_maths
Satisfying 🫧 #integral #integration #maths #stem #calculus
#Sumation Reel by @navamienterprises2614 - The integration of \(e^{-x^{2}}\) (the Gaussian function) is considered one of the "hardest" or most famous non-elementary integrals because it has no
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@navamienterprises2614
The integration of \(e^{-x^{2}}\) (the Gaussian function) is considered one of the "hardest" or most famous non-elementary integrals because it has no antiderivative that can be expressed in terms of elementary functions. Mathematics Stack Exchange +1While the function \(f(x)=e^{-x^{2}}\) is perfectly smooth and continuous—meaning it is definitely integrable in the sense that an area under the curve exists—you cannot define its indefinite integral using standard functions like polynomials, exponentials, logarithms, or trigonometric functions. Mathematics Stack Exchange +1Here is why it is uniquely difficult: 1. No "Elementary" Antiderivative Most calculus integration techniques (like substitution or integration by parts) are designed to reverse-engineer differentiation. However, there is no elementary function \(F(x)\) that you can differentiate to get \(e^{-x^{2}}\). This has been formally proven using Liouville's theorem, which dictates that certain elementary functions do not have The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function  f(x)=e−x2  over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is  ∫−∞∞e−x2dx=π.
#Sumation Reel by @mathswithmuza - A sum and an integral are both ways of accumulating quantities, but they operate in slightly different settings. A sum adds up a finite or countable c
237.6K
MA
@mathswithmuza
A sum and an integral are both ways of accumulating quantities, but they operate in slightly different settings. A sum adds up a finite or countable collection of discrete values, like adding the areas of several rectangles or summing terms in a sequence. For example, when you compute a Riemann sum, you divide an interval into pieces and add up the areas of rectangles under a curve. Integration takes this idea to the limit. Instead of adding a fixed number of rectangles, you let the number of pieces grow while their widths shrink toward zero. The definite integral represents the exact accumulated quantity that those increasingly refined sums are approaching. So conceptually, an integral is the limit of sums. The difference becomes especially important in continuous versus discrete models. A sum is appropriate when you are working with distinct data points, such as values of a sequence or probabilities of discrete outcomes. An integral is used when quantities vary continuously, such as computing area under a smooth curve, total distance from a velocity function, or accumulated mass from a density function. In Fourier analysis, for instance, Fourier series use sums because they combine countably many sine and cosine terms, while the Fourier transform uses an integral because it blends together a continuous range of frequencies. Both tools measure accumulation, but the choice between them depends on whether the underlying structure is discrete or continuous. Like this video and follow @mathswithmuza for more! #math #integral #physics #foryou #animation
#Sumation Reel by @equationacademy - ➡️Visualizing Limit of Sinc Function ➡️ Follow @equationacademy for more #mathematics #calculus #technology #fyp #science
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@equationacademy
➡️Visualizing Limit of Sinc Function ➡️ Follow @equationacademy for more #mathematics #calculus #technology #fyp #science

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