Calc Integral: |sinx-cos2x| from 0 to \pi/2

In summary, the purpose of calculating this integral is to find the area under the curve of the function |sinx-cos2x| from x=0 to x=π/2. This can be done using integration techniques such as substitution or integration by parts. The absolute value in the function indicates that the area can be both positive and negative, making it important to carefully consider the behavior of the function in different intervals. This integral can be solved analytically, but the resulting solution may need to be evaluated numerically. Real-world applications of this integral include calculating work in physics and determining average prices in economics.
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[tex]\int^{\pi/2}_{0}|sinx-cos2x| dx[/tex]
Hi guys i can't seem to get the correct answer for this. What i did was draw the graph of sinx and cos2x and for 0 to [tex]\pi/6[/tex] i use cos2x-sinx since cos2x is higher. From [tex]\pi/6[/tex] to [tex]\pi/2[/tex] i use sinx-cos2x. However i am not able to get the same answer as my book can someone tell me is this the way to do integrals involving the absolute value?
 
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  • #2
You've split the integral up correctly. Your problem must be in evaluating the two integrals.
 

1. What is the purpose of calculating this integral?

The purpose of calculating this integral is to find the area under the curve of the function |sinx-cos2x| from x=0 to x=π/2. This can be useful in various applications such as finding the average value of a function or determining the total change of a variable over a given range.

2. How do you approach solving this integral?

To solve this integral, you can use integration techniques such as substitution or integration by parts. It is important to carefully identify the limits of integration and break down the function into simpler components to make the integration process easier.

3. What is the significance of the absolute value in the function?

The absolute value in the function |sinx-cos2x| indicates that the area under the curve can be both positive and negative. This means that the function is not always increasing or decreasing, and the total area can be affected by the behavior of the function in different intervals.

4. Can this integral be solved analytically?

Yes, this integral can be solved analytically using integration techniques. However, the resulting integral may not have a closed-form solution and may need to be evaluated numerically using software or techniques such as Simpson's rule.

5. What are some real-world applications of this integral?

This integral can be used in various fields such as physics, engineering, and economics. For example, in physics, it can be used to calculate the work done by a force that changes with respect to position. In economics, it can be used to determine the average price of a product over a specific time period.

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