WEBVTT

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Hi, I'm Fiona.

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Let's have a look at Tamua 2021 paper one
and question 2.

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We're given a curve y = X ^3 -,
6 X +3 and told that it has turning

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points at X equals alpha and X equals
beta, with beta being greater than alpha.

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And we're asked to integrate this curve
between alpha and beta.

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We're given some options to choose from
once we get to that point,

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and I'm going to start this one simply by
finding alpha and beta.

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And to do that,
I'm going to differentiate so DYDX equals

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three X ^2 -, 6.

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Now I'm going to set that equal to 0 and
that will give me the X values at which

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this function has turning points,
which will give me alpha and beta.

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So three X ^2 - 6 = 0,
which means that three X ^2 = 6,

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which means X ^2 = 2.

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And that means that X is going to equal
plus or minus sqrt 2.

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Now I'm told that beta is bigger than
alpha.

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That means that alpha is minus root 2 and
beta is a positive root 2.

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So this means that I'm going to be
integrating between minus root 2 and a

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positive root 2.

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So to save space I'm just going to do
that.

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And now we simply just integrate and so
this integration,

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the first term will become X to the power
of 4 / 4,

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the second term will be X ^2 and then
that will be a -, 3 X squared.

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And then this third term will be plus 3X.

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And I am evaluating this definite
integral between minus root 2 and root 2,

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and so first just substituting in root 2.

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I have root 2 all raised to the power of
4 / 4 - 3 root 2 ^2 + 3 times root 2,

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and I'm going to take away minus root 2
to the power of 4 / 4 -.

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3 times minus root 2 ^2 + 3 times minus
root 2.

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So that's -3 root 2.

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And what I can see here is that because
I'm taking away these three terms from

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these three terms,
I can see that this term take away this

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term is going to be 0.

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This term take away this term is going to
be 0 because the powers on root 2 and

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minus root 2 are even and so I'm just
going to be left with three root 2 -, -,

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3, root 2.

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So I'm just going to be left with three
root 2 + 3 root 2,

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which is going to be 6 times root 2.

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Looking at my options,
I can see that this is option F So the

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answer to this question is option F6 root
2.