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

TL;DR

The video explains how to evaluate a definite integral using the power rule, with an example calculation included.

Transcript

• [Voiceover] So, we want to evaluate the definite integral from negative one to eight of 12 times the cube root of x dx. Let's see, this is going to be the same thing as the definite integral from negative one to eight of 12 times, the cube root is the same thing as saying x to the 1/3 power dx and so now, if we want to take the antiderivative of ... Read More

Key Insights

• ☺️ The definite integral is used to find the total area between a function and the x-axis within a specified interval.
• ✊ The power rule for integrals is used to find the antiderivative of a power function.
• 🥡 The value of a definite integral represents the net area between the function and the x-axis, taking into account the direction of the curve above and below the x-axis.
• ❓ The definite integral can be evaluated by finding the antiderivative and applying the bounds of integration.
• ✊ Integration is the reverse process of differentiation, and the power rule is the reverse of the power rule for derivatives.
• 👻 The definite integral allows for the computation of areas, volumes, and accumulation of quantities.
• ✊ The power rule is a useful tool for evaluating integrals involving power functions.

Questions & Answers

Q: How is the definite integral of 12 times the cube root of x evaluated from -1 to 8?

The definite integral is evaluated by finding the antiderivative of x to the 1/3 power using the power rule. The antiderivative is then substituted into the definite integral and evaluated at the upper and lower bounds, subtracting the value at the lower bound from the value at the upper bound.

Q: What is the antiderivative of x to the 1/3 power?

The antiderivative of x to the 1/3 power is found using the power rule for integrals. The exponent is increased by one and then divided by the increased exponent. In this case, it becomes x to the 4/3 power divided by 4/3.

Q: How are the bounds -1 and 8 used in the evaluation of the definite integral?

The expression, nine x to the 4/3 power, obtained from the antiderivative, is evaluated at 8 and -1. The value at -1 is subtracted from the value at 8 to find the result of the definite integral.

Q: What is the final answer for the definite integral from -1 to 8 of 12 times the cube root of x?

The final answer is 135.

Summary & Key Takeaways

• The video demonstrates how to evaluate the definite integral of 12 times the cube root of x from -1 to 8.

• The antiderivative of x to the 1/3 power is found using the power rule, and the definite integral is rewritten in terms of this antiderivative.

• The expression is evaluated at the bounds, subtracting the value at -1 from the value at 8, resulting in an answer of 135.