Find Upper Limit of Integral: "Integration Please Help?

In summary, to find the upper limit of an integral, you must identify the function and lower limit, and then use techniques such as substitution or integration by parts. A calculator can be used but may not always give the exact answer. The purpose of finding the upper limit is to determine the area under a curve or the value of a function within a specific range. Helpful tips include understanding the problem, identifying patterns, and practicing regularly. The fundamental theorem of calculus can also be used to find the upper limit.
  • #1
MarkFL
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Here is the question:

Integration please help?

f(x)=2sinx

a is somewhere between 0 and π
the area between f(x), a=x and axis equals 1
find a?

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello jasna,

We are being asked to solve the following equation for $a$:

\(\displaystyle 2\int_0^a \sin(x)\,dx=1\)

Divide through by $-2$:

\(\displaystyle \int_0^a -\sin(x)\,dx=-\frac{1}{2}\)

On the left, apply the FTOC:

\(\displaystyle \left[\cos(x) \right]_0^a=-\frac{1}{2}\)

\(\displaystyle \cos(a)-1=-\frac{1}{2}\)

\(\displaystyle \cos(a)=\frac{1}{2}\)

\(\displaystyle a=\frac{\pi}{3}\)
 

FAQ: Find Upper Limit of Integral: "Integration Please Help?

How do I find the upper limit of an integral?

To find the upper limit of an integral, you need to first identify the function being integrated and the lower limit of the integral. Then, you can use various techniques such as substitution, u-substitution, or integration by parts to solve for the upper limit. It is important to carefully follow the steps and make sure to correctly evaluate the integral to get the correct upper limit.

Can I use a calculator to find the upper limit of an integral?

Yes, you can use a calculator to find the upper limit of an integral. However, it is important to note that calculators can only provide numerical approximations and may not always give the exact answer. It is always best to double-check your work and use a calculator as a tool to assist in the process.

What is the purpose of finding the upper limit of an integral?

The upper limit of an integral is used to determine the area under a curve or the value of a function within a specific range. It is a fundamental concept in calculus and is used in various fields such as physics, engineering, and economics to solve real-world problems.

Are there any tips for finding the upper limit of an integral?

One helpful tip for finding the upper limit of an integral is to carefully read the problem and make sure you understand what is being asked. Also, try to identify any patterns or relationships within the function that can help simplify the integration process. It is also important to practice regularly to become more comfortable with different integration techniques.

Can I use the fundamental theorem of calculus to find the upper limit of an integral?

Yes, the fundamental theorem of calculus can be used to find the upper limit of an integral. This theorem states that the upper limit of an integral can be found by evaluating the antiderivative of the function at the upper limit and subtracting the antiderivative at the lower limit. This method is often used to simplify the integration process and can be especially useful for complex functions.

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