1. ## Double Integrals

There are two of them that I can't figure out:

1. $\int^2_0\int^{\frac{\pi}{2}}_0 x\sin{y} \ dydx$

On this one I got 0, but it's supposed to be 2

And

2. $\int^2_0\int^1_0 (2x + y)^8 \ dxdy$

On this one I got $\frac{262144}{45}$ but its supposed to be $\frac{261632}{45}$. My answer is $2^9$ off.

Any help is appreciated.

2. $\int^2_0\int^{\frac{\pi}{2}}_0 x\sin{y} \ dydx$

$\int^2_0 x~dx \int^{\frac{\pi}{2}}_0 \sin y~dy$

I think you can do it now.

3. $\int_{0}^{2}\int_{0}^{\frac{\pi}{2}}xsin(y)dydx$

The first integration yields:

$\int xsin(y)dy=-xcos(y)$

Now, using the limits:

$-xcos(\frac{\pi}{2})+xcos(0)=x$

Now for that part:

$\int_{0}^{2}x dx$

Of course, this is easy enough:

$\frac{1}{2}(2)^{2}-\frac{1}{2}(0)^{2}=\boxed{2}$

4. Thanks. Didn't consider the cos(0).

Thanks wingless... I forgot about splitting the functions.

Anyone for the second problem?

5. Make the substitution $z=2x+y$ for the inner integral, the rest is just straightforward computation.

6. Originally Posted by Aryth
Thanks. Didn't consider the cos(0).

Thanks wingless... I forgot about splitting the functions.

Anyone for the second problem?
the second one we can do by substitution.

first letting u = 2x + y, we get: $\int_0^2 \frac 1{18} (2x + y)^9 \bigg|_0^1 ~dy$

continue in the same manner. was that what you tried?

7. Ah, I did it the same way, but I see what I did wrong... It was in the first iteration:

$\frac{1}{18}(2x + y)^9\bigg|_0^1 = \frac{1}{18}\left[(2 + y)^9 - {\color{red}{y^9}}\right]$

That would completely explain why I was $2^9$ over. I completely forgot the lower limit.

8. Originally Posted by Aryth
Ah, I did it the same way, but I see what I did wrong... It was in the first iteration:

$\frac{1}{18}(2x + y)^9\bigg|_0^1 = \frac{1}{18}\left[(2 + y)^9 - {\color{red}{y^9}}\right]$

That would completely explain why I was $2^9$ over.
ah, ok, good catching your mistake

9. Thanks... Bad news is, I made the same mistake on both problems. I gotta watch the lower limits. :P

10. Originally Posted by Aryth
Thanks... Bad news is, I made the same mistake on both problems. I gotta watch the lower limits. :P
haha, indeed. now you know you weakness, you can work at squashing it. this should not hurt you in an exam now