Name: definite integral rules for motion
Uploaded: 2026-01-12T12:24:01.000Z
Duration: 1 min 17 s
Channel: JoVE

定積分は移動した変位のような量を計算します。異なる速度で移動する2台の車を比較する際は、積分の正性と比較特性が用いられます。車の速度v(t)を時間区間[a,b]で考えます。車が後ろに動かないため、速度 v(t) は [a,b] で非負です。定積分は移動した変位を示します。関数が非負であるため、積分の値も非負であり、積分の正の性質を示しています。次に、速度関数f(t)とg(t)を持つ2台の車、車1と車2が同じ区間を移動していると想像します。もしf(t)が任意の時刻tでg(t)以上であれば、車1は車2に対してより大きいか等しい変位を移動しなければなりません。したがって、時間区間 [a,b] における f(t) の定積分は g(t) の定積分以上であり、積分の比較性質を示しています。

定積分の性質 iii

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定積分の性質 III

Education

JoVE Core

Calculus

微積分学

Unit

単位

Integrals

積分

KEY TERMS AND CONCEPTS

主要用語と概念

Properties of Definite Integral III

definite integral rules for motion

The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations.

The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the object does not move backward. For example, if a car keeps moving forward or stays still during a trip, the distance it covers cannot be negative. The definite integral adds up the forward motion and gives a nonnegative result.

The Comparison Property lets us compare the motion of two objects over the same time interval. If Car 1 always moves at a speed equal to or greater than Car 2, then Car 1 travels at least as far as Car 2. This property shows that a consistently faster object has greater or equal accumulated displacement. The definite integral captures that difference by summing each velocity over time.

Together, these properties make integrals more useful for understanding physical systems. They show how a math tool can measure accumulated change in motion. They also help students connect the abstract idea of integration with real-world travel.

Definite Integral Rules for Motion

area problem

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Approximating area under a curve is a key way to measure regions with curved boundaries. Straight-sided shapes are easy to handle with formulas for rectangles, triangles, and polygons. Curved regions need a different method because standard geometry does not work well on them. The area problem solves this by breaking the region into smaller, simpler shapes. A common approach uses rectangles to estimate the space under a function. Each rectangle gets its height from the function at a chosen...

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sums of power

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Riemann sums use rectangles to estimate the area under a curve in definite integration. The interval is split into subintervals, and the rectangle areas are added together. When those rectangle heights follow a regular pattern, sum formulas can make the calculation faster and more exact. This page focuses on sums that appear in arithmetic and polynomial sequences. The sum of consecutive integers, squares, and cubes is especially useful when the partition is uniform or when the data has a...

midpoint rule

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properties of definite integral i

Definite Integrals for Car Motion

Definite integrals help describe a car’s motion over time by linking velocity to displacement. In a velocity-time graph, the definite integral shows how velocity builds up over a chosen time interval. The area under the curve gives the total displacement, so the graph becomes a clear picture of how far the car moves. One key idea is linearity of definite integrals. This property lets you add or subtract velocity functions while keeping their accumulated effect. If a car’s motion is made of...

properties of definite integral ii

Definite Integrals in Motion Analysis

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points. Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent...

properties of definite integral iii

The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations. The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the...

fundamental theorem of calculus i

Definite Integrals: A Step-by-Step Strategy

Definite integrals are solved best with a clear, step-by-step strategy. The first step is to identify what the problem is asking for. The quantity may be accumulation, area, force, or probability. It is also important to note the integration limits and any constraints that affect the setup. Before evaluating the integral, look for properties that can simplify the work. If the expression has multiple parts, break it into smaller integrals. Known values from earlier work may help too. Symmetry...

fundamental theorem of calculus i: problem solving

fundamental-theorem-of-calculus-i-problem-solving

Accumulation and Rates in the Fundamental Theorem

The Fundamental Theorem of Calculus connects accumulation and rate of change. In this lesson, the idea is shown with a water tank that is filled by a pump. The pump rate changes over time, so the total water collected must be found by adding up the changing inflow. In water management, this kind of calculation matters for pressure control, operation timing, and system safety. The flow rate is given by a time-dependent function. To find how much water has entered the tank from the start to a...

fundamental theorem of calculus ii

Definite Integrals with Antiderivatives

The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative. If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because...

application of integration: problem solving

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Integration in Breathing Cycle Modeling

Integration can be used to model the volume of air inhaled during breathing. The respiratory cycle is the repeating intake and expulsion of air, and it usually lasts about five seconds. By treating airflow as a function of time, students can see how the amount of air in the lungs changes during a breath. Airflow during respiration is not constant. It changes in a smooth, wave-like pattern, so a sinusoidal function is a good model. In a normal breathing cycle, the airflow rate reaches a maximum...

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Finding Water Volume with Integration

Integration can be used to find the total volume of water in a tank when the inflow rate changes over time. In this example, the pump starts at 5 L/min. The inflow rate then increases by 2 L/min for each additional minute. That changing rate can be written as a linear function. To measure how much water has been added, the inflow rate function is integrated over time. The total water volume V(t) comes from this integral. When the power rule is used, the result includes both a quadratic term...

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Calculus for Reservoir Overflow Prediction

The Net Change Theorem links a function’s rate of change to its total accumulated change over an interval. In calculus, this means the definite integral of a derivative on [a,b] gives the net change in the original function. The theorem is useful in physics, economics, and engineering because it turns a rate into a total amount. One clear application is a reservoir that receives water from two inflow sources at the same time. The reservoir starts with 260 million cubic meters of water. Each...

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substitution-rule-applied-to-indefinite-integrals

Work Done on a Nonlinear Spring

Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward. A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown,...

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substitution-rule-applied-to-definite-integrals

Using Substitution to Evaluate Definite Integrals

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Finding Fish Biomass with Integrals

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key terms and concepts

Learn how the Positivity Property and Comparison Property of definite integrals describe motion, velocity, and displacement in real-world examples.

Definite Integral Rules for Motion | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Definite Integral Rules for Motion

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