What Is the Integral of |sin(x)| from 0 to 3pi/2?

TL;DR

The integral of the absolute value of sine x from 0 to 3π/2 evaluates to 3. This is determined by splitting the integral into two parts due to the sine function's positive and negative regions and applying the derivative of the cosine function in the calculations.

Transcript

integrate the absolute value of sine x from zero to three PI over two so whenever you have an integral with the absolute value you should always try to get rid of the absolute value so the definition of the absolute value of x is a piecewise function it's equal to X if X is greater than or equal to zero and it's equal to minus X if X is less than z... Read More

Key Insights

• 🤝 The absolute value of a function can be defined using piecewise functions when dealing with integrals.
• ❎ Analyzing the positive and negative regions of a function is useful for splitting the integral.
• 👨‍💼 The derivative of the cosine function is negative sine x, which can be used in evaluating the integral of sine x.
• 😑 Evaluating definite integrals requires substituting the limits and simplifying the expression.
• ❓ Understanding the properties and behavior of trigonometric functions is fundamental in calculus.
• 👻 Splitting the integral based on the sign of the function allows for easier integration.
• ☺️ The integral of the absolute value of sine x yields a positive value.

Questions & Answers

Q: How is the absolute value of sine x defined?

The absolute value of sine x is equal to sine x when sine x is non-negative and equal to -sin x when sine x is negative.

Q: How can the integral of the absolute value of sine x be split into two parts?

By analyzing the positive and negative regions of the sine function, the integral is split from 0 to π and from π to 3π/2.

Q: What is the derivative of the cosine function?

The derivative of the cosine function is negative sine x, so when integrating sine x, it becomes positive cosine x.

Q: How is the integral evaluated to get the final result of 3?

Evaluating the integral using the derivative of the cosine function, the result is obtained by plugging in the limits and simplifying the expression, which gives a final answer of 3.

Summary & Key Takeaways

• The absolute value of sine x is defined as sine x when sine x is greater than or equal to zero, and as -sin x when sine x is less than zero.

• By analyzing the positive and negative regions of the sine function, the integral can be split into two parts: one from 0 to π and another from π to 3π/2.

• Evaluating the integral using the derivative of the cosine function, the final result is found to be 3.