Use series to approximate the definite integral i to within...

Use series to approximate the definite integral $I$ to within the indicated accuracy. $I = \int_{0}^{0.5} x^4 e^{-x^2} dx$ ($|error| < 0.001$) $I = \boxed{}$ You may have rounded in the wrong direction.

Transcript

0:00 So suppose we want to do this.

00:01 We want to do 0 .2 .5 x squared, right? i want to do this one, dx.

00:22 What is e to the x? e to the x, the x, the summation, x, that, right? so what is e to the negative that? when we see x, we're going to put negative x squared.

00:38 So it is going to be negative x squared.

00:41 So multiplied by n is going to be negative x squared.

00:42 To be negative x to 2 n right over n factorial now this negative one here can be pulled out so this can be negative 1 to the n bring our estimation then we have our x to the 2n over n factorial right now we have to multiply by this so when you multiply by x squared what is happening this is going to be when you multiply this one by x squared and actually you're going to add right so that's what we're going to do so when you do that then it's going to be summation in from zero to infinity negative one -half to the and x to the two and plus one or n factorial that is what is happening right so once you have this one then we can now do the integration right so when you do the integration is 0 to 0 .5 so 0 to 0 .5 0 .2 .5, right, the x.

01:47 So the integration is going to just affect this guy right here, right? so when it affects that one, then you're going to have summation negative 1 to the n.

02:01 Now this is going to be x to 2 n plus 2 over 2 in plus 2.

02:07 Do not forget your n factorial, right? and you evaluating it at 0 and 0 .5, right? the zero part is going to go away because when you put x equals zero here it's going to go so finally what you're going to have is uh this guy right here right point five to the power 2 n plus 2 over 2 n plus 2 in factorial right that is that is this this guy right here good so once you have this all you have to do is evaluate right uh so when n is zero what is when n is zero, you're going to have, you know, we're trying to do it at three decimal places accuracy, right? so we need when in is zero, what is happening? so when in is zero, so when we multiply it here by x squared, let me correct something a little here.

03:17 We multiplied this one by x squared isn't it where is yeah x squared so when we multiplied it by x squared we multiplied here by x squared so it's going to be 2n plus 2 right so this is supposed to be 2 n plus 2 right so uh claim this one and then we do here as integral this is x squared e to the negative x squared d x right that is that is that is that is uh what is to be so that is a little correction here.

03:52 So once you have that, and then this integral here is going to be plus three now, right? it's not plus two.

03:57 It's going to be plus three now.

03:59 So this is plus three, plus three, and it's going to continue from here, plus three, right? so this is plus three, plus three.

04:10 Good.

04:10 So now you're going to follow the same thing.

04:12 So when in is zero, what are we going to get? because we're now, this is an alternating series...

Question

Use series to approximate the definite integral $I$ to within the indicated accuracy. $I = \int_{0}^{0.5} x^4 e^{-x^2} dx$ ($|error| < 0.001$) $I = \boxed{}$ You may have rounded in the wrong direction.

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