Fundamental Theorem of Calculus: Changing the Bounds with u-Substitution

turksvids
6 Jan 201903:18
EducationalLearning
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TLDRThe video presents a unique approach to evaluating a definite integral using substitution and changing the bounds of integration. The integral in question is from negative 2 to 4 of the function (x/(2+1))^3. The presenter chooses 'u' as the expression inside the parentheses and finds that du = 1/2 dx, leading to the substitution of 2 du = dx. By evaluating 'u' at the bounds -2 and 4, the new bounds are determined to be 0 and 3, respectively. The integral is then transformed into an integral in terms of 'u', resulting in 2∫(u^3) du from 0 to 3. The integral is solved by applying the power rule in reverse, yielding (1/4)u^4. The final answer is obtained by evaluating this expression at the new bounds, resulting in (81/2), which is the value of the original definite integral. The video effectively demonstrates the process of substitution and the importance of correctly changing the bounds when evaluating integrals.

Takeaways
  • 📚 The video discusses an example of the Fundamental Theorem of Calculus with a substitution that changes the bounds.
  • 🔍 The integral to be evaluated is from -2 to 4 of (x / (2 + 1))^3.
  • 🎯 A substitution is chosen because it simplifies the integral by avoiding the need to cube the term and then reverse the power rule.
  • 👉 The substitution is set with u = (x / 2) + 1, which implies du = (1/2) dx.
  • 🧐 The bounds for the integral are changed to reflect the new variable u, with u(-2) = 0 and u(4) = 3.
  • 🔄 The integral is rewritten in terms of u, with the bounds now from 0 to 3.
  • 📉 The integral in terms of u is simplified by taking out the constant 2, resulting in 2 * integral of u^3 du.
  • 🧮 The power rule is reversed to solve the new integral, which gives (1/4)u^4.
  • ⏫ The new integral is evaluated from the new bounds, resulting in (1/4) * (3^4 - 0^4).
  • 🔢 The final answer is calculated as 81/2, which is the value of the original definite integral from -2 to 4.
  • 📈 The process demonstrates the utility of substitution in calculus to simplify and solve complex integrals.
Q & A

  • What is the main topic of the video?

    -The video is about demonstrating an example of the Fundamental Theorem of Calculus, specifically by changing the bounds after performing a substitution.

  • What is the integral the video aims to evaluate?

    -The video aims to evaluate the integral from negative 2 to 4 of the function (x / (2 + 1))^3.

  • Why does the video suggest using a substitution?

    -The video suggests using a substitution because the integral looks complex and cubing the term would make it more difficult to solve without reversing the power rule.

  • What is the substitution variable 'u' chosen to be?

    -The substitution variable 'u' is chosen to be x / (2 + 1), which simplifies to x / 2 + 1/2.

  • How does the video change the bounds of the integral after the substitution?

    -The video changes the bounds by evaluating 'u' at the original bounds (-2 and 4), resulting in new bounds of 0 and 3, respectively.

  • What is the new integral in terms of 'u'?

    -The new integral in terms of 'u' is ∫(2du) * u^3, where the integral is evaluated from the new bounds 0 to 3.

  • How does the video simplify the integral after the substitution?

    -The video simplifies the integral by factoring out the constant 2 and then applying the power rule in reverse to integrate u^3, resulting in (1/4)u^4.

  • What are the new bounds for the integral in terms of 'u'?

    -The new bounds for the integral in terms of 'u' are from 0 to 3.

  • How does the video evaluate the new integral?

    -The video evaluates the new integral by plugging in the bounds into the simplified expression, which gives (1/2) * (3^4 - 0^4).

  • What is the final answer for the definite integral?

    -The final answer for the definite integral is 81/2.

  • Why is it important to remember the original integral when evaluating the equivalent integral?

    -It is important to remember the original integral to ensure that the substitution and the evaluation of the equivalent integral accurately represent the original problem and yield the correct result.

  • What does the video imply about the relationship between the original and the equivalent integral?

    -The video implies that the original and the equivalent integral are equivalent in value, as demonstrated by the process of substitution and evaluation, which should yield the same result.

Outlines
00:00
🧮 Applying U-Substitution in Definite Integrals

The video begins by introducing a calculus problem involving a definite integral with a complex integrand, x/(2+1)^3, from -2 to 4. The presenter decides to apply a substitution method (u-substitution) to simplify the integral. They choose u to be the inner expression (x/(2+1)) and find du to be 1/2 dx, which is then rewritten as 2 du = dx. The bounds of integration are then transformed using the substitution, changing from -2 and 4 to 0 and 3, respectively. The integral is then rewritten in terms of u, resulting in an integral of u^3. The power rule is applied in reverse to integrate u^3, yielding u^4/4. The integral is evaluated from the new bounds, resulting in (3^4/4) - (0^4/4), which simplifies to 81/2. The presenter concludes by emphasizing that the value of the original integral is indeed 81/2, demonstrating the effectiveness of the substitution method.

Mindmap
Keywords
💡Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a foundational result that connects differentiation and integration, two of the main operations in calculus. In the video, it is used to evaluate a definite integral by transforming it into an indefinite integral and then applying the theorem to find the difference between the antiderivative evaluated at the bounds.
💡Substitution
Substitution is a technique used in calculus to simplify the evaluation of integrals. By substituting a variable, the integrand can be transformed into a simpler form. In the video, the substitution method is used to change the bounds of the integral after identifying a suitable substitution variable, 'u', to simplify the integral.
💡Definite Integral
A definite integral is an integral that has a specific lower and upper bound, representing the area under a curve between these two points. In the video, the definite integral from negative 2 to 4 of a given function is evaluated using substitution and the Fundamental Theorem of Calculus.
💡Indefinite Integral
An indefinite integral is an integral without specific bounds and is used to find the antiderivative of a function. It is the reverse process of differentiation. In the video, the process of finding an indefinite integral is demonstrated by reversing the power rule after substitution.
💡Power Rule
The power rule is a basic rule in calculus that allows for the differentiation of polynomial functions. It states that the derivative of x^n, where n is a constant, is n*x^(n-1). In the video, the power rule is mentioned in the context of reversing it to integrate a function.
💡Bounds
Bounds in the context of an integral refer to the limits between which the integral is evaluated. In the video, the bounds are initially given as negative 2 and 4, but they are changed after performing a substitution to reflect the new limits in terms of the substitution variable 'u'.
💡Antiderivative
An antiderivative is a function whose derivative is the original function. Finding an antiderivative is a process known as integration. In the video, the antiderivative of the function 'x over (2 + 1) cubed' is found by applying the power rule in reverse after substitution.
💡Cubed Function
A cubed function is a function where the variable is raised to the power of three. In the video, the function 'x over (2 + 1) cubed' is simplified using substitution to avoid the complexity of cubing the entire expression.
💡Reversing Power Rule
Reversing the power rule is a technique used in integration to find the antiderivative of a function. It involves applying the power rule in the opposite direction to find the original function. In the video, this technique is used to integrate 'u cubed' by finding 'u to the fourth' and then evaluating it at the new bounds.
💡Integration by Substitution
Integration by substitution is a method used to evaluate integrals that are difficult to solve directly. It involves changing the variable to make the integral easier to solve. In the video, the method is demonstrated by choosing 'u' as the substitution variable to transform the integral and then evaluating the new integral in terms of 'u'.
💡Change of Bounds
Change of bounds is a technique used when performing substitution in integration. It involves finding the new limits for the integral after the substitution has been made. In the video, the bounds are changed from the original 'negative 2 to 4' to '0 to 3' after substituting 'u' for 'x over (2 + 1)'.
Highlights

The video demonstrates an example of using substitution to evaluate a definite integral with a change of bounds.

The integral to be evaluated is ∫ from -2 to 4 of (x / (2 + 1))^3 dx.

The approach is to perform a substitution to simplify the integral before evaluating it.

Let u = (x / (2 + 1)), which simplifies the integral after substitution.

Then rewrite the integral in terms of u: 2 du = dx.

Change the bounds from x = -2 to x = 4 to u = 0 to u = 3.

The new integral in terms of u is ∫ from 0 to 3 of u^3 * 2 du.

Simplify the integral by factoring out the 2: 2 * ∫ from 0 to 3 of u^3 du.

Apply the power rule to evaluate the new integral in terms of u.

The result is (1/4)u^4, evaluated from 0 to 3.

Plug in the bounds to get (3^4 - 0^4)/2 = 81/2.

The final answer is 81/2 for the original integral from -2 to 4.

The technique of substitution followed by a change of bounds is demonstrated.

The approach simplifies the integral by avoiding having to cube the expression inside the integral.

The substitution makes it easier to apply the power rule to evaluate the integral.

Changing the bounds to match the new variable u is a key step in the process.

Factoring out constants like 2 from the integral simplifies the evaluation.

Applying the power rule directly to the simplified integral in terms of u is straightforward.

The final answer is obtained by plugging in the bounds and simplifying.

The video provides a clear step-by-step example of using substitution to evaluate a definite integral.

The method is useful for integrals that are difficult to evaluate directly due to complex expressions inside the integral.

Transcripts

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