Question

Use a taylor series to approximate definite integral of sin(x^2)dx from x=0 to x=0.2

Answer 1

fo you know series for sine?

Answer 2

yes the series is x^2 - x^6/6+x^10/120-x^14/5040+...

Answer 3

replace \(x\) in that series for \(x^2\) an then take it out as far as you care to
series is
\[x-\frac{x^3}{3!}+\frac{x^5}{5!}-...\] replace \(x\) by \(x^2\) get
\[x^2-\frac{x^6}{3!}+\frac{x^{10}}{5!}-...\]

Answer 4

just replace \(x\) by \(x^2\) not the denominators

Answer 5

Yes so I got that part right. thank you. how do I write the sigma notation ?

Answer 6

since you are close to zero, you can probably get a good estimation by taking
\[\int_0^{.2}x^2dx\]

Answer 7

the sixth power term will not contribute that much

Answer 8

thanks