Question

Calculus III double Integrals....
How do you get to the last step of this picture?

Answer 1

dxdy = r dr d (theta)

Answer 2

the steps are given to you already, what do you not understand about it?

Answer 3

They skipped a few explanations in there, maybe this will help a little bit.

\[\large z=f(x,y)\]

\[\large f(x,y)=11-\sqrt{1+x^2+y^2}\]Converting to polar using these identities:\[\large \color{green}{x=r \cos \theta} \qquad \qquad \qquad \color{purple}{y=r \sin \theta}\]

Changing our function to,\[\large f(r,\theta)=11-\sqrt{1+(\color{green}{r \cos \theta})^2+(\color{purple}{r \sin \theta})^2}\]
Which will simplify down a bit,
\[\large f(r,\theta)=11-\sqrt{1+r^2(\color{orangered}{\cos^2\theta+\sin^2\theta})}\]
Recalling another important identity,\[\large \color{orangered}{\cos^2\theta+\sin^2\theta=1}\]

So our function simplifies to,\[\large f(r,\theta)=11-\sqrt{1+r^2}\]

Answer 4

But as Comp asked, it would be nice if we knew what particular spot you're stuck on.

Answer 5

the part you just solved was the part i was stuck on... The final step, did not know how the r before the square root disappeared.

Answer 6

The r before the square root, oh ok.
Let's look at that part of the integral,

\[\large \int\limits r (1-r^2)^{1/2}\; dr\]

Performing a `U-substitution`:\[\large \color{royalblue}{u=1+r^2}\]Taking the derivative gives us,\[\large du=-2r\;dr \qquad\rightarrow\qquad \color{red}{-\frac{1}{2}du=r\;dr}\]

We'll substitute these into our integral giving us,\[\large \int\limits\limits (\color{royalblue}{1-r^2})^{1/2}\color{red}{r\; dr} \qquad\rightarrow\qquad \int\limits\limits (\color{royalblue}{u})^{1/2}\left(\color{red}{-\frac{1}{2}du}\right)\]

Answer 7

To integrate this, we simply apply the `Power Rule for Integrals`

\[\large -\frac{1}{2}\int\limits u^{1/2}du \qquad=\qquad -\frac{1}{2}\frac{u^{3/2}}{3/2} \qquad=\qquad -\frac{1}{3}u^{3/2}\]

Answer 8

u substitution, i figured as much...

Answer 9

i do not do well with u substitution at all....

Answer 10

mm yah they're a tad tricky :D

Answer 11

where does -1/2r come from?

Answer 12

−1/2du=rdr