Definite integral of radical function | AP Calculus AB | Khan Academy

Khan Academy
28 Jul 201603:50
EducationalLearning
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TLDRThe video script presents a step-by-step explanation of evaluating a definite integral of a function from negative one to eight. The process involves applying the power rule for integration, which involves increasing the exponent by one and dividing by the new exponent. The integral is of 12 times the cube root of x, and the antiderivative is found to be 12 times x to the 4/3 power divided by 4/3. The final result is obtained by evaluating this expression at the upper bound (eight) and subtracting the value at the lower bound (negative one), leading to a final answer of 135.

Takeaways
  • 📌 The definite integral is evaluated from negative one to eight of 12 times the cube root of x.
  • 🔢 The integral can be rewritten as ∫(12x^(1/3)) dx, where x^(1/3) represents the cube root of x.
  • 📚 Applying the power rule for integration, which is the reverse of the power rule for differentiation, is essential for solving the integral.
  • 🧠 The power rule states that the integral of x^n is (x^(n+1))/(n+1) + C, where n is the exponent and C is the constant of integration.
  • 🌟 The antiderivative of 12x^(1/3) is 12x^(4/3) / (4/3), simplifying to 9x^(4/3) / (4/3).
  • 🎯 To find the definite integral, evaluate the antiderivative at the upper bound (8) and subtract the value at the lower bound (-1).
  • 🔍 Evaluating the antiderivative at the upper bound gives 9 * (8^(4/3)), which simplifies to 9 * 16.
  • 📈 Evaluating the antiderivative at the lower bound gives 9 * (-1)^(4/3), which simplifies to 9 * 1 because (-1)^4 = 1.
  • 🧮 The final result is 9 * (16 - 1), which equals 9 * 15, and the definite integral is 135.
  • 📊 This problem demonstrates the process of evaluating definite integrals using the power rule and the importance of carefully applying the bounds of integration.
Q & A

  • What is the integral being evaluated in the script?

    -The integral being evaluated is the definite integral from negative one to eight of 12 times the cube root of x.

  • How is the cube root of x expressed differently for the integration process?

    -The cube root of x is expressed as x to the 1/3 power for the integration process.

  • What mathematical rule is applied to find the antiderivative in the integration process?

    -The power rule for integrals is applied, which involves increasing the exponent by one and dividing by that new exponent.

  • What is the new exponent of x when applying the power rule for integrals to x to the 1/3?

    -The new exponent of x is 4/3, obtained by adding 1 to 1/3.

  • How is the integral simplified before evaluating it at the bounds?

    -The integral is simplified by multiplying 12 by the reciprocal of 4/3, which simplifies to 9 times x to the 4/3 power.

  • What are the steps to simplify 12 divided by 4/3?

    -12 divided by 4/3 is simplified by multiplying 12 by the reciprocal of 4/3, resulting in 12 times 3/4, which equals 9.

  • How is 8 to the 4/3 power calculated and what is its value?

    -8 to the 4/3 power is calculated by taking the cube root of 8 (which is 2) and then raising it to the fourth power, resulting in 16.

  • What is the value of negative one to the 4/3 power and how is it derived?

    -Negative one to the 4/3 power is calculated as one, by first raising negative one to the fourth power to get one, and then taking the cube root of that, which is still one.

  • What is the final result of evaluating the definite integral and how is it obtained?

    -The final result of the definite integral is 135, obtained by evaluating the expression 9 times x to the 4/3 power at the bounds 8 and -1, and then simplifying.

  • Why does the script mention simplifying and evaluating the expression in different colors?

    -Mentioning the use of different colors suggests that the script is visually distinguishing between various steps of the calculation process for clarity and instructional purposes.

Outlines
00:00
🧮 Solving a Definite Integral Involving a Cube Root

The video explains the process of evaluating a definite integral of the function 12 times the cube root of x, across the interval from -1 to 8. It begins by converting the cube root into a fractional exponent, leading to the expression 12x^(1/3). Applying the reverse power rule for integrals, the exponent is increased by one (resulting in 4/3) and the function is divided by this new exponent. Simplification yields a coefficient of 9 times x^(4/3). The solution involves evaluating this expression at the bounds 8 and -1, simplifying using exponent properties, and finally calculating the difference to obtain the integral's value of 135.

Mindmap
Keywords
💡Definite Integral
A definite integral is a fundamental concept in calculus that represents the area under a curve of a function over a specified interval. In the video, the definite integral is used to calculate the area under the curve of the function 12 times the cube root of x from -1 to 8. The process involves finding the antiderivative of the function and then evaluating it at the bounds of the interval.
💡Cube Root
The cube root of a number is a value that, when cubed, gives the original number. In the context of the video, the cube root of x is part of the function being integrated. The process of integrating this function involves applying the power rule, which is a fundamental technique in calculus for integrating functions involving exponents.
💡Antiderivative
An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function. In the video, finding the antiderivative is a step in evaluating the definite integral. The antiderivative is found by applying calculus rules, such as the power rule, to the integrand.
💡Power Rule
The power rule is a basic rule in calculus that states if you have a function of the form x^n, where n is any real number, then its derivative is n*x^(n-1). In the context of the video, the power rule is used both for differentiation and integration. For integration, the rule is applied in reverse, increasing the exponent by one and dividing by the new exponent.
💡Bounds
Bounds refer to the limits of the interval over which an integral is calculated. In the video, the bounds are the numbers -1 and 8, which define the interval from -1 to 8 on the x-axis. The definite integral's value is found by evaluating the antiderivative at these bounds and subtracting the lower bound value from the upper bound value.
💡Exponent Properties
Exponent properties are the rules that govern how exponents behave when they are combined or manipulated. These properties include the multiplication of powers, the power of a power, and the division of powers. In the video, exponent properties are used to simplify expressions, particularly when evaluating the antiderivative at the bounds.
💡Evaluation
Evaluation in the context of the video refers to the process of substituting the bounds into the antiderivative to find the value of the definite integral. It involves simplifying the expression and performing the necessary arithmetic to arrive at the final numerical value.
💡Area Under a Curve
The area under a curve is a geometric concept that represents the size of the region enclosed by the curve and the x-axis. In the video, the definite integral is used to calculate this area for the specified function over the given interval. This is a practical application of integration, which has many uses in fields such as physics, engineering, and economics.
💡Calculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. It consists of two main subfields: differential calculus, which deals with derivatives, and integral calculus, which deals with integrals. The video is focused on integral calculus, specifically the concept of definite integrals and their application in calculating areas under curves.
💡Simplification
Simplification in mathematics refers to the process of making a mathematical expression or equation easier to understand or calculate by reducing it to a simpler form. In the video, simplification is used to make the calculation of the definite integral more manageable by reducing complex expressions to simpler ones.
Highlights

Evaluating a definite integral involving a cube root function

Integrating from negative one to eight of 12 times the cube root of x

Applying the power rule for integrals, which is the reverse of the power rule for derivatives

Increasing the exponent by one and dividing by the new exponent

The antiderivative is 12 times x to the power of 4/3

Evaluating the antiderivative at the bounds of the integral

Subtracting the value of the antiderivative at negative one from the value at eight

Simplifying the expression by finding the value of 12 divided by 4/3

Expressing the result as 9 times x to the power of 4/3

Calculating eight to the 4/3 power using exponent properties

Finding that eight to the 4/3 is equal to 16

Determining negative one to the 4/3 power using exponent rules

Calculating that negative one to the 4/3 is equal to one

The final result is nine times 15, which equals 135

The process demonstrates a clear and methodical approach to solving integrals involving power functions

The explanation is detailed, providing a step-by-step guide for understanding the integral calculus concepts

Transcripts

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MathematicsIntegralsPower RuleCube RootAntiderivativeCalculusEducationMath TutorialProblem SolvingMathematics Education