Simpson's Rule derived from Lagrange Interpolation. Confused, please help.

Question

I'm reading my lecturer's notes on how to derive the Simpson's Rule using Lagrange's Interpolating Polynomial, but there's a point that doesn't quite seem right.

Here's a screenshot of the notes pointing out where I'm confused:

My problem is, how is it true that:

$$ \int_{-h}^{h} {(x+h)x \over (2h)(h)} y_2 = {{y_2} \over 2h^2 } \left[ {x^3 \over 3} - {hx^2 \over 2} \right]_{-h}^{h} $$

Where did the $ \space - {hx^2 \over 2} \space $ come from?

Shouldn't it be:

$$ \int_{-h}^{h} {(x+h)x \over (2h)(h)} y_2 = {{y_2} \over 2h^2 } \left[ {x^3 \over 3} + {hx^2 \over 2} \right]_{-h}^{h} $$

And if that's the case, doesn't the whole proof break?

Sorry if this is easy and straightforward to some, I'm really lost here.

Looks like a copy and paste error from the the first term. The rest appears okay on first sight — user251257, Oct 17 '17 at 18:02
@user251257 - Ahh right, so the rest onward is right aside from that slight error? (I really should check it myself, I'm very sorry) — Max Echendu, Oct 17 '17 at 18:11
@MaxEchendu don’t worry. That happens pretty often, as many rules are symmetric and people like copy&paste — user251257, Oct 17 '17 at 18:13

score 3 · Accepted Answer · answered Oct 17 '17 at 18:12

3

You're exactly correct. The last term of the integral (in your first red box) is expressed correctly, but evaluated incorrectly (in your second red box).

answered Oct 17 '17 at 18:12

John

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Simpson's Rule derived from Lagrange Interpolation. Confused, please help.

1 Answers1