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Double integrals 2

Petrus

Well-known member
Feb 21, 2013
739
Hello MHB,
I would like to have tips how to solve the x limits for this problem
,

there



Regards,
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
Hello MHB,
I would like to have tips how to solve the x limits for this problem
,
there



Regards,
Your area D is bounded by the 2 curves $y=x^6$ and $y=x^{1/5}$.
Where do they intersect?
Or put differently, can you solve that set of 2 equations?
 

Petrus

Well-known member
Feb 21, 2013
739
Your area D is bounded by the 2 curves $y=x^6$ and $y=x^{1/5}$.
Where do they intersect?
Or put differently, can you solve that set of 2 equations?
hmm \(\displaystyle x^6=x^{1/5} <=> x_1=1 x_2=0\)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774

Petrus

Well-known member
Feb 21, 2013
739
Yep.
Those are the x limits in your problem.
Hello I like Serena,
I forgot to say thanks, I solved it!:)

For anyone who is intrested how to solve it:
\(\displaystyle \int_0^1\int_{x^6}^{x^{\frac{1}{5}}} x^4y^6 \ dydx\)
We can always use Fubini's theorem and take out the constant and integrate the y so we got

\(\displaystyle \int_0^1 x^4 \left[\frac{y^7}{7}\right]_{x^6}^{x^{\frac{1}{5}}}\)
\(\displaystyle \int_0^1x^4\frac{x^{\frac{7}{5}}}{7}-x^4\frac{x^{42}}{7} dx\)
use the rule \(\displaystyle x^m*x^n=x^{n+m}\) and we can simplify to ( we can take out a constant as well):

\(\displaystyle \frac{1}{7}\int_0^1x^{\frac{27}{5}}-x^{46} dx\)
\(\displaystyle \frac{1}{7} \left[ \frac{5x^{\frac{32}{5}}}{32} - \frac{x^{47}}{47} \right]_0^1 = \frac{29}{1504}\)

Regards,