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As you have learned this semester, there are many different techniques of integration designed to
handle different types of functions and situations As you have seen. some are harder than others.
You have not seen every technique available and there are some problems that have no
antiderivatives at all! Those complexities cause trouble when we have an application problem
that needs use to solve a definite integral, and integration is the first step. Fortunately we have
power series as they can be of immense help in integration
For the problems below, you must show all work for credit. You may freely use the power series
you have learned for er sin(x)and cos(x). Any other series for any other specific function
must be built in your work You must do your own work on your own paper and upload that to
the appropriate dropbox in cCampus no later than 6 AM on Wednesday June 1. The work must
be clear and legible. If I cannot read it or decipher it I will not grade it.
I. One simple function that has no algebraic antiderivative is y=d" Power series are a
good tool to help solve integral applications with that function Set up the definite
integral to find the volume for the solid of revolution for the region between - and
the x-axis over revolved on the x-axis. Then find a closed form power series for
the function you are actually integrating. Using the first 6 non-zero terms of that series,
calculate an approximation for the actual volume form the solid of revolution The
answer, accurate to 8 significant digits is 7.42815098. How close were you?
2. Now we will calculate the area inside the polar curve r - 205(30)sin(30).
First, build a definite integral to represent this area Calculate the value of this area
exactly using integration techniques from the course this semester. Second, find the first
6 non-zero terms for the power series for the integrand; there are many ways to do this
but you must justify your answer however you get it. Then use that approximation to
find an approximate value for your definite integral. How close to your exact value did
you get? If you wanted to improve your approximation what would you do?

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