Calculating Iterated Integrals - 2e4

[Kork]

1. The problem statement, all variables and given/known data

Calculate the given iterated integrals ∫₀² dy ∫₀^yy² * e^xy dxMy attempt:

∫²₀dy[e^xy*y]^y₀

= ∫²₀ e^y*y*y - e^x*0*0

= ∫²₀e^y2*y dx

= [e^{y^2}]*y]²₀ = 2e⁴

Is this correct?

 

[Mark44]

1. The problem statement, all variables and given/known data

Calculate the given iterated integrals ∫₀² dy ∫₀^yy² * e^xy dx

I think you skipped a step going from the above to the line below. Written in a more usual form, your integral would be
\int_{y = 0}^2~\int_{x = 0}^y y^2 e^{xy}~dx~dy

(You can see what I did in LaTeX by clicking the integral above.)
Show how you get from the step above to the step where you've carried out the inner integration.

My attempt:

∫²₀dy[e^xy*y]^y₀

= ∫²₀ e^y*y*y - e^x*0*0

= ∫²₀e^y2*y dx

= [e^{y^2}]*y]²₀ = 2e⁴

Is this correct?

No - see above. I get (1/2)e⁴ - 5/2

 

[Kork]

I don't understand what I have done wrong at all

 

[Mark44]

The error is here:
\left . ye^{xy}\right |_{x = 0}^y

You need to replace x by 0, not y.

 

[Kork]

Im still confused I get:

[e^{y^2}*y-y]²₀ =

(e^2^2 - 2) - (e^0^2 - 0) =

2e^4 - 2 ?

Im lost

 

[Mark44]

You're skipping steps. You have the outer integrand right, but you have made a mistake when you integrated this integral.
\int_0^2 (ye^{y^2} - y)dy

Split this into two integrals and carry out the two integrations.

 

[Kork]

Oh I have totally lost my comprehensive view now...

 

[Mark44]

Oh I have totally lost my comprehensive view now...

I don't know what you mean by this.

 

[Kork]

How do I get from:

= ∫(y = 0 to 2) (ye^(y^2) - y) dy

= (1/2)e^(y^2) - (1/2)y^2 {for y = 0 to 2}

?

 

[Kork]

∫_y⁰ ye^(y^2) - y dy

to

= (1/2)e^(y^2) - (1/2)y^2


{for y = 0 to 2}

 

[Mark44]

Split the integral into two.
Use substitution to do the first integral.

 

[Kork]

This didnt get me further from start, but thanks anyway.

 

[Mark44]

∫_y⁰ ye^(y^2) - y dy

to

= (1/2)e^(y^2) - (1/2)y^2

When you evaluate the above at 2 and 0, what do you get?

Sorry, I misunderstood what you were asking before.

{for y = 0 to 2}