With this definite integral, we can find its antiderivative and find the area.

Let's take a look at the graph...

Solving problems like this are not always this easy as this one. We can't always find the exact value of the definite integral.

Take a look at this problem and its graph.

We could use Riemann sums to find the area under the curve.

____________________________________________________________________

These are the differentiation and integration formulas. We use these to help us determine the problems below.

Determine:

1)

Use substitution. (I FORGOT TO PUT dx INTO THE PICTURE)

u= x

^{3}

du= 3x

^{2}

1/3du= x

^{2}dx

If your wondering why u = x

^{3}its because

(x

^{3})

^{2}= x

^{6}. So u is u

^{2}. We are trying to make the original function look like d/dx Arctan(x) integration formula.

^{}

Substitute ....................................................Resubstitute

You don't need " + C " in the intermediate steps because the C are not the same while your substituting.

(There's supposed to be an equal (=) sign in between the pictures)

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2)

If the 8 was a 9 in the polynomial we could complete the square.

So we do this,

6x-x

^{2}-8 Factor out the negative.

-[(x

^{2}-6x+8 +1) -1]

-[(x

^{2}-6x+9) -1]

-[(x-3)

^{2}-1]

1 - (x-3)

^{2}We can replace this into the integral.

Next we use substitution.

Let u = x+3

du = dx

Then RE-substitute..

And we're done

Homework: finish up arctrig exercises (7.5)

Wednesday's scribe will be Katrin.

## 2 comments:

Hi Ashlynn,

I have a question about #1 if you don't mind. You said:

u= x 3

du= 3x2

1/3du= x2 dx

but your integral has only xdx. Is it a typo or am I not seeing something obvious? Thanks,

e

Yes that is a typo. Thank you for telling me.

The picture was supposed to read X squared.

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