Definite integral with substitution

Khan Academy
2 Jul 200808:54
EducationalLearning
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TLDRThe video script presents a detailed walkthrough of solving a definite integral, specifically from pi over 2 to pi, of the function minus cosine squared of x times sin of x. The explanation begins with a graphical representation of the function to establish an intuitive understanding of the negative area under the curve that the integral represents. The solution process is systematically broken down, first using the concept of substitution with a clear explanation of the choice of substitution variable (u as cosine of x), and then by applying the reverse chain rule to reinforce the concept. The final answer, minus 1/3, is derived and verified with both methods, demonstrating their equivalence and providing viewers with alternative approaches to tackling similar problems.

Takeaways
  • 📈 The problem involves evaluating a definite integral of a trigonometric function from pi/2 to pi.
  • 🤔 The key is to understand the concept of definite integrals as the area between the curve and the x-axis.
  • 📊 The graph of the function -cos(x)^2 * sin(x) is below the x-axis, indicating the integral's value will be negative.
  • 🧠 Intuition suggests that the result of the integral should be a negative number, which is confirmed by the graph.
  • 🔄 The process starts by rearranging the terms of the integral to make it easier to work with.
  • 📐 Substitution is a useful method for solving integrals involving products of functions and their derivatives.
  • 🌀 By setting u = cos(x), the derivative du/dx = -sin(x)dx allows for a straightforward integration by parts.
  • 🔄 After substitution, the integral simplifies to u^2 du from 0 to -1, which is easier to evaluate.
  • 📈 The antiderivative of u^2 is (u^3)/3, and by evaluating this at the bounds, the result is -1/3.
  • 🔄 An alternative method mentioned is using the reverse chain rule, which is quicker but essentially the same as substitution.
  • 🎯 Both methods lead to the same result, confirming the integral's value is -1/3, which matches the area interpretation.
Q & A

  • What is the integral problem presented in the script?

    -The integral problem presented is the evaluation of the definite integral from pi over 2 to pi of minus cosine squared of x times sin of x dx.

  • What does the definite integral represent in this context?

    -In this context, the definite integral represents the area of the curve between the function and the x-axis from pi over 2 to pi.

  • Why is the area calculated from the curve in this problem expected to be negative?

    -The area is expected to be negative because the curve is below the x-axis between pi over 2 and pi, indicating a negative region.

  • What substitution method was suggested for solving the integral?

    -The substitution method suggested involves setting u equal to the cosine of x, which allows the integral to be solved by evaluating the antiderivative of u squared from 0 to -1.

  • How is the derivative of cosine of x used in the substitution method?

    -The derivative of cosine of x, which is minus sin of x, is used to express du/dx. This allows for the substitution of u for cosine of x and du for minus sin of x dx in the integral.

  • What is the antiderivative of u squared found using the substitution method?

    -The antiderivative of u squared is u cubed over 3, which is found by increasing the exponent by one and dividing by the new exponent.

  • What is the final result of the definite integral?

    -The final result of the definite integral is minus 1/3, representing the negative area under the curve between pi over 2 and pi.

  • How does the script also mention an alternative method to solving the integral?

    -The script mentions an alternative method using the reverse chain rule, which is a quicker approach that treats the cosine of x term as an x or u term for finding the antiderivative without formal substitution.

  • What is the significance of the graph in understanding the integral problem?

    -The graph provides an intuitive understanding of the integral by visually showing the area under the curve that needs to be calculated, helping to confirm the expected sign and magnitude of the result.

  • How does the script emphasize the importance of understanding the problem before solving it?

    -The script emphasizes the importance of understanding the problem by encouraging the viewer to think about what the definite integral represents and to develop an intuition for the expected result before proceeding with the mathematical solution.

Outlines
00:00
📚 Introduction to Definite Integral Problem

The video begins with the presenter introducing a definite integral problem involving the function minus cosine squared of x times sin of x, from pi over 2 to pi. The presenter emphasizes the importance of understanding the problem by looking at examples and visualizing the function through a graph. The integral represents the area between the curve and the x-axis, which in this case is negative due to the curve being below the x-axis. The presenter then proceeds to solve the integral using the fundamental theorem of calculus, considering the use of substitution or the reverse chain rule for simplification.

05:00
🔢 Solution and Intuition of the Integral

In this paragraph, the presenter dives deeper into solving the integral problem. They rearrange the terms for clarity and decide to use substitution to find the antiderivative. By setting u as the cosine of x, the derivative du/dx is found to be minus sin of x, which simplifies the integral to u squared du. The boundaries are then evaluated with u at cosine of pi over 2 and cosine of pi. The integral is then rewritten as the antiderivative of u squared, which is u cubed over 3. After evaluating the antiderivative at the boundaries, the result is minus 1/3, which matches the negative area under the curve as visualized in the graph. The presenter also briefly explains an alternative method using the reverse chain rule for a quicker solution, reaffirming the correctness of the result obtained through substitution.

Mindmap
Keywords
💡Definite Integral
A definite integral represents the signed area under a curve between two points on the x-axis. In the video, the definite integral is calculated for the function -cos^2(x)*sin(x) from pi/2 to pi, which corresponds to the area under the curve but also takes into account where the curve lies relative to the x-axis.
💡Cosine (cos)
Cosine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of its adjacent side to its hypotenuse. In the video, the function -cos^2(x) represents the square of the cosine function, which is then used in the definite integral.
💡Sine (sin)
Sine is another fundamental trigonometric function, similar to cosine, that relates the angle of a right triangle to the ratio of its opposite side to its hypotenuse. In the context of the video, sin(x) is a component of the integrand and also the derivative of cos(x), which is crucial for the substitution method used to solve the integral.
💡Substitution
Substitution is a technique used in calculus to simplify the process of integration by replacing a part of the integrand with a new variable. In the video, the substitution method is used by setting u as cos(x), which allows for the simplification and evaluation of the integral.
💡Chain Rule
The chain rule is a fundamental calculus technique used to find the derivative of a composite function. It states that the derivative of a function of a function is the derivative of the outer function times the derivative of the inner function. Although not directly used to solve the integral in the video, the concept is mentioned as a method to deal with composite functions.
💡Antiderivative
An antiderivative is a function whose derivative is the given function we are trying to integrate. In the process of integration, finding the antiderivative of the integrand allows us to apply the fundamental theorem of calculus to evaluate the definite integral.
💡Fundamental Theorem of Calculus
The fundamental theorem of calculus provides a relationship between differentiation and integration. It states that if a function is continuous on a closed interval [a, b] and F is an antiderivative of f on that interval, then the definite integral of f from a to b is equal to F(b) - F(a).
💡Graphing Function
Graphing a function involves visually representing the relationship between variables on a coordinate plane. In the context of the video, graphing the function -cos^2(x)*sin(x) helps to visualize the area under consideration for the definite integral.
💡Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved. They are used to simplify and manipulate trigonometric expressions. In the video, the identity cos^2(x) is implicitly used when discussing the integrand and the substitution method.
💡Negative Area
Negative area refers to the region under a curve that lies below the x-axis on a graph. In the context of definite integrals, a negative area indicates that the contribution to the integral is subtracted from the total area.
💡Evaluation
Evaluation in the context of definite integrals refers to the process of finding the value of the integral by substituting the limits of integration into the antiderivative. It is the final step in calculating the net area under the curve between the given interval.
Highlights

The integral problem involves the function minus cosine squared of x times sin of x from pi over 2 to pi.

The key to solving this problem is understanding the definite integral and its representation of the area between the curve and the x-axis.

The graph of the function minus cosine squared of x times sin of x is shown, with the area of interest being below the x-axis, indicating a negative area.

The problem is approached by rearranging terms and considering the use of substitution or the reverse chain rule for solving the integral.

The chain rule is mentioned as a guideline for finding the derivative of a composite function, which may be useful in this problem.

A substitution is made with u equal to cosine of x, and the derivative du/dx is found to be minus sin of x.

The integral is rewritten with the substitution, transforming the original function into u squared times du.

The boundaries for the substitution are calculated, with u going from cosine of pi over 2 to cosine of pi.

The integral is simplified to the antiderivative of u squared, which is u cubed over 3.

The antiderivative is evaluated at the boundaries, resulting in minus 1/3, which represents the area under the curve.

The negative area is confirmed by the graph, and the absolute area is found to be 1/3.

An alternative method using the reverse chain rule is introduced, which can be quicker for solving such problems.

The reverse chain rule approach treats the cosine of x term as an x term for finding the antiderivative.

The final answer using the reverse chain rule method is the same as the substitution method, which is minus 1/3.

The problem-solving process provides an intuitive understanding of the area under the curve and the application of calculus concepts.

The demonstration of two methods for solving the integral problem showcases the flexibility and power of calculus techniques.

Transcripts

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MathematicsIntegralsCalculusDefinite IntegralCosine FunctionSine FunctionSubstitution MethodReverse Chain RuleProblem SolvingEducational Content