The prompt for all of these question is "consider the function f(x) = sin^2(x)".

Question

Part A: Write the first four terms of the Maclaurin series for f(x).

I assumed this implied non-zero terms, so I found x^2-(1/3)x^4+(2/45)x^6-1/315(x^8). I am fairly confident about this.

Part B: Use the Maclaurin series found in Part A to approximate the integral from 0 to 1 of sin^2(x)

I know sin^2(x) is about equal to the Maclaurin series around x=0, so the integrated Maclaurin series would be (x^3)/3-(x^5)/15+2(x^7)/315-(x^9)/2835. Evaluated an an integral from 0 to 1 yields 0.2726631, an answer I am confident in.

Part C: How many terms are needed to find the value of the integral given in Part B, correct to three places? What is that value?

I am unsure what "correct to three decimal places" means exactly, because most questions typically give an actual value as a error bound. I am also unsure whether to consider the original terms of the Maclaurin series, or the anti-derivatised terms that will be used to approximate the integral. For example I know that the fourth non-zero term of the integrated series will be ((1)^9)/2835 = 0.0003527, which seems to be "correct to three decimal places", but I am not sure if three zeros is what is implied by that. If I look at non integrated terms, then the fifth term is the first to have three zeros 2((x)^10)/14175 = 0.0001411.

Overall I am just confused on whether I should be looking at integrated terms of the Maclaurin series, or the original terms, as well as what meant by "correct to three places". Any help would be greatly appreciated!

Answer 1

Part A:
There are different ways to do this part, but your answer is correct. Hope you used the simplest way.

Part B:
Again, you answer is correct. Using integration term by term is also correct.

Part C:
"correct to 3 decimal places" generally means that the sum of the remaining (discarded) terms must not be greater than 0.0005. In an alternating series (as in this case), it is sufficient to have the first discarded term less than 0.0005.

The above procedure applies to the series of the final answer, i.e. the integrated terms, so your procedure is correct.
The 3-term approximation is 0.2730 and the exact integral is 0.2727, both of which are as predicted in your analysis.

Having said that, we must note that the procedure above does not guarantee that the third digit after the decimal is correct. For example, if the correct answer is 0.245499, and the approximation is 0.245623. The error is only 0.000124, but the rounded values are respectively 0.245 and 0.246.