Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skillsβperfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding the Laplace Transform and Its Purpose
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we will discuss how the Laplace Transform simplifies calculations involving integrals. Can anyone tell me what the Laplace Transform is?
It's a method to transform a function from the time domain to the s-domain, right?
Exactly! And why do we do this? It helps us solve differential equations more easily. Now, who can give me the definition of the Laplace Transform?
It's defined as L{f(t)} = β«_0^β e^(-st) f(t) dt.
Perfect! Todayβs focus will be on using this transform to solve integrals. Let's talk about what happens when you integrate a function and then take its Laplace Transform. Can anyone tell me about the relationship?
I think itβs related by dividing the original transform by s.
Correct! That leads us to our theorem. Remember: when we have an integral of a function, L{β«_0^t f(Ο) dΟ} = F(s)/s. Let's explore a practical example to see this in action.
Example Problem 1: Integral of sin(aΟ)
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Letβs look at the first problem. We need to find L{β«_0^t sin(aΟ) dΟ}. What should we do first?
We need to find the Laplace Transform of sin(at) first?
Exactly, so what's L{sin(at)}?
Itβs a/sΒ² + aΒ².
Great! Now applying our theorem: L{β«_0^t sin(aΟ) dΟ} = F(s)/s. Can someone apply it?
So, L{β«_0^t sin(aΟ) dΟ} = a/{s * (sΒ² + aΒ²)}.
Excellent work! This was an application of our theorem. Now letβs summarize.
Example Problem 2: Integral of e^{2Ο}
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Moving on, we have the next example: L{β«_0^t e^{2Ο} dΟ}. Whatβs f(t) here?
f(t) is e^{2t}.
And how do we find its Laplace Transform?
The Laplace Transform L{e^{2t}} is 1/(s - 2) for s > 2.
Correct! Now, applying our theorem again, what do we get?
We get L{β«_0^t e^{2Ο} dΟ} = 1/{s * (s - 2)}.
Exactly! Excellent job! We see the usefulness of this approach in simplifying integrals. Let's recap what we've learned today.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, two example problems illustrate the process of finding the Laplace Transform of integrals using the theorem established for such cases. These examples serve to reinforce the theoretical framework discussed earlier in the chapter by applying it in practical scenarios.
Detailed
Example Problems
This section provides two example problems that utilize the theorem on the Laplace Transform of integrals, reinforcing the understanding of how this technique simplifies the evaluation of integrals in engineering mathematics.
Key Concept Overview
The Laplace Transform is defined for a function f(t) and transforms it to F(s). Specifically, when we have an integral of the form g(t) = β«_0^t f(Ο) dΟ, the theorem states that:
$$L\{g(t)\} = \frac{F(s)}{s}$$
where F(s) is the Laplace Transform of the original function f(t). This lends itself well to the analysis of systems described by such integrals, widely used in electrical and control systems engineering.
Example 1 Explanation
In the first example, we find the Laplace Transform of the integral of the sine function:
$$L\{\int_0^t \sin(aΟ) dΟ\}$$
Here, we define f(t) = sin(at) leading to the transformation leveraging the theorem.
Example 2 Explanation
In the second example, we handle an exponential function:
$$L\{\int_0^t e^{2Ο} dΟ\}$$
Defining f(t) = e^{2t} gives us another practical application of the theorem. These exercises not only illustrate the theorem but also highlight its utility in solving related engineering problems.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Example 1: Laplace Transform of Sine Integral
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 1:
Find the Laplace Transform of
\[ g(t) = \int_0^t \sin(a\tau) d\tau \]
Solution: Let f(t)=sin(at), then
\[ F(s)=L\{\sin(at)\}=\frac{a}{s^2 + a^2} \]
Using the theorem,
\[ L\{\int_0^t \sin(a\tau) d\tau\} = \frac{F(s)}{s} = \frac{a}{s(s^2 + a^2)} \]
Detailed Explanation
In this example, we need to find the Laplace transform of the integral of the sine function. We denote the integral as g(t) which represents the accumulated area under the curve defined by sin(aΟ). According to our initial setup, the Laplace Transform of sin(at) gives us F(s). Applying the theorem allows us to express the Laplace Transform of the integral in terms of F(s). By substituting F(s) into our equation, we derive the final result that shows how the Laplace Transform of the integral can be simplified.
Examples & Analogies
Think about filling a tank with water where the rate of water flowing in follows a sine wave pattern. The integral represents the total amount of water in the tank at time t. By calculating the Laplace Transform, we can analyze how the tank behaves over time, despite the changing rate of flow.
Example 2: Laplace Transform of Exponential Integral
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 2:
Find \( L\{ \int_0^t e^{2\tau} d\tau \} \)
Solution: Let f(t)=e^{2t} \Rightarrow F(s)= \frac{1}{s-2}, s>2
So,
\[ L\{\int_0^t e^{2\tau} d\tau \} = \frac{1}{s(s-2)} \]
Detailed Explanation
In this second example, we want to find the Laplace Transform of another type of integral, specifically one involving an exponential function. We start by expressing f(t) as e^(2t) and then compute its Laplace Transform to obtain F(s). Using the given theorem again allows us to relate the Laplace Transform of the integral directly to the calculated F(s). This demonstrates how different functions can be transformed similarly using the same theoretical approach.
Examples & Analogies
Imagine a bank account where money is continuously added at an exponential rate (representing e^(2Ο)). The integral calculates the total amount over time, which might represent your total balance at any given moment. The Laplace Transform helps us model this accumulation in a useful mathematical form, allowing us to predict future balances based on current rates.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Laplace Transform: A mathematical operation that transforms a time domain function into a frequency domain function.
-
Integral of a Function: The process of calculating the area under the curve of a function over a specified interval.
-
Piecewise Continuous Functions: Functions that are continuous within the limits of certain intervals but may have jump discontinuities.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1: Finding L{β«_0^t sin(aΟ) dΟ results in a/(s * (sΒ² + aΒ²)).
-
Example 2: Finding L{β«_0^t e^{2Ο} dΟ results in 1/(s * (s - 2)).
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
-
For sin and cos, we integrate with ease, Remember 'F over s', it's like a breeze!
π Fascinating Stories
-
Imagine a mathematician named Sam who loves transformations. One day he realized that when he integrated fun functions, they could be simplified by 'dividing by s', and he felt as if he had discovered a new treasure in the land of Laplace.
π§ Other Memory Gems
-
I.S.E. β Integrate, Simplify, Evaluate for remembering the order for Laplace Transforms involving integrals.
π― Super Acronyms
F.I.S.H. - Functions Integrate Simplistically Hence β to recall that F(s) is divided by s for integrals.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Laplace Transform
Definition:
A technique that transforms a time-domain function into a frequency domain representation, helping to solve differential equations.
-
Term: Integral
Definition:
A mathematical operation that calculates the area under a curve defined by a function, over a specified interval.
-
Term: Piecewise Continuous Function
Definition:
A function that is continuous on a series of intervals but may have a finite number of discontinuities.
-
Term: Exponential Order
Definition:
A property of a function whereby it grows no faster than an exponential function as its variable approaches infinity.