Calculus
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The Role of Integrals in Solving Differential Equations
Differential equations are at the core of mathematical modeling. They describe how things change. This could be the motion of a pendulum. It might also refer to the growth of a population. Moreover, it could mean the cooling of a metal rod. But identifying the equation is only part of the task. The real challenge…
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Unbounded limits
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Understanding Improper Integrals in Calculus
In calculus, integration is often introduced as a way to measure area under a curve. For most functions and intervals, this is straightforward. But not all mathematical problems are that well-behaved. Some functions spike to infinity. Others stretch endlessly across the number line. Standard integrals can’t handle these cases. Improper integrals can. What Are Improper…
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Integrals: Tools for Measuring Economic Surplus
In economics, the value of a transaction is not measured solely by the price paid. Behind every market exchange lies a surplus, value gained by both consumers and producers that goes beyond the monetary cost. Understanding these surpluses is critical to evaluating the efficiency and fairness of markets. Integrals, although mathematical in origin, serve as…
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Volume of Solids: Disk, Washer, and Shell Methods
In calculus, you often move from working with flat, two-dimensional shapes to exploring the three-dimensional objects they can create. One fascinating way this happens is by rotating a flat shape around an axis, forming a solid of revolution. To determine the volume that these solids occupy, integration is used. It’s a method of adding up…
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Calculating Areas with Definite Integrals
Mathematics helps you to understand the world in powerful ways, and one of its key tools is calculus. Among its many applications, finding the area under a curve using definite integrals is an essential concept with real-world significance. This method is versatile, applicable to subjects like physics and engineering. It also extends to economics and…
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Gravitational potential energy – quite short
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Still struggling with the concept of Integration—watch this!
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Simplifying Integrals: Substitution & Parts Methods
Integration is a fundamental concept in calculus, allowing you to find areas, accumulated quantities, and solutions to real-world problems. While some integrals are straightforward, others require special techniques to solve efficiently. Two of the most important methods are substitution and integration by parts. These techniques help simplify complex problems, making integration more manageable. Substitution: Changing…
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Fundamental theorem of calculus, how it works (Animation) – short