Integration by parts question: int [x^3 g"(x)] = x^3g'(x) - int [3x^2g'(x)]

[Karim]

Hi! I'm stuck on the problem below.
I attempted to use integration by parts but got stuck on the following:



The integral(3x^2g'(x)) must be further integrated because it's two unique functions so should I continue using integration by parts on this part till I get a single function?

 

[BigBeachBanana]

Hi! I'm stuck on the problem below.
I attempted to use integration by parts but got stuck on the following:


View attachment 34951
The integral(3x^2g'(x)) must be further integrated because it's two unique functions so should I continue using integration by parts on this part till I get a single function?

View attachment 34950

The problem statement says [imath]g'(x)[/imath], not [imath]g''(x)[/imath]. Try again.

 

[Karim]

Yes, I'm aware
But I'm using integration by parts so I use the format
integral(u * v') = (u * v) - integral(u' * v)

 

[BigBeachBanana]

Yes, I'm aware
But I'm using integration by parts so I use the format
integral(u * v') = (u * v) - integral(u' * v)

From the problem statement, what's your [imath]u[/imath] and [imath]v'?[/imath]

 

[Karim]

Ok I get what you're saying and used that method but I'm still stuck at the part integral(3x^2g(x)). What method should I use to integrate this part?

 

[BigBeachBanana]

Ok I get what you're saying and used that method but I'm still stuck at the part integral(3x^2g(x)). What method should I use to integrate this part?


View attachment 34953

That's part 2 of the question. Use the left Riemann Sum to approximate.

 

[Karim]

If I write it as this my u = x^3 and v' = g'(x)


If I write it as this my u = x^3 and v' = g''(x)

 

[Karim]

Ohh so I can't further simplify beyond this point?

 

[BigBeachBanana]

This is correct but you need the differentials for your integrals [imath]dx's[/imath]


I don't understand why you keep changing to [imath]g''(x)[/imath] when the problem statement says [imath]g'(x)[/imath].

Ohh so I can't further simplify beyond this point?

As I said above, use Riemann Sum to approximate as instructed.

 

[Karim]

Ok so I got the following, will the next step give the final answer?

 

[BigBeachBanana]

Ok so I got the following, will the next step give the final answer?
View attachment 34957

Show your work for the Riemann Sum. I didn't get the same answer.

[math]\int_0^3x^3g'(x)~ dx = x^3g(x) \bigg\rvert_{0}^{3} - \int_{0}^{3}3x^2g(x)~ dx \\[/math][math]\int_0^3x^3g'(x)~ dx \approx x^3g(x) \bigg\rvert_{0}^{3} - \sum_{i=1}^{6}3x_i^2\, g(x_i)~ \Delta x[/math]

 

[BigBeachBanana]

This is my working
View attachment 34962
Forgot to write that the areas were being multiplied by 0.5 (the width of the Reinmann sum rectangle), the final answer takes the width into account

Did you use a calculator? The calculations are incorrect. For example [imath]3(1)^2 \times 3.7[/imath] can't be [imath]5.55[/imath] because we know [imath]3(3) = 9[/imath].

 

[Karim]

Hi, so I was able to fix my calculations. Did I get the same answer as you?

 

[BigBeachBanana]

Hi, so I was able to fix my calculations. Did I get the same answer as you?

View attachment 34964

Off a bit by rounding, but the steps are correct. I got 80.28.

 

[Steven G]

Did you use a calculator? The calculations are incorrect. For example [imath]3(1)^2 \times 3.7[/imath] can't be [imath]5.55[/imath] because we know [imath]3(3) = 9[/imath].

There is a (0.5) in the front of 3(1)2×3.73(1)2×3.7
Corner time!

 

[limiTS]

I used a spreadsheet and got 80.4