MHB 15.3 Express an integral for finding the area of region bounded by:

[karush]

ok so there are 3 peices to this
Express and integral for finding the area of region bounded by:

\begin{align*}\displaystyle
y&=2\sqrt{x}\\
3y&=x\\
y&=x-2
\end{align*}

View attachment 7250

 

[skeeter]

Re: 15.3 Express and integral for finding the area of region bounded by:

ok so there are 3 peices to this
Express and integral for finding the area of region bounded by:

\begin{align*}\displaystyle
y&=2\sqrt{x}\\
3y&=x\\
y&=x-2
\end{align*}

http://mathhelpboards.com/attachments/calculus-10/7250-15-3-express-integral-finding-area-region-bounded-15-3-png

$3y=x \implies y = \dfrac{x}{3}$

first intersection ...

$x-2 = \dfrac{x}{3} \implies x = 3$

second intersection ...

$x-2 = 2\sqrt{x} \implies x=4+2\sqrt{3}$
$\displaystyle A = \int_0^3 2\sqrt{x} - \dfrac{x}{3} \, dx + \int_3^{4+2\sqrt{3}} 2\sqrt{x} - (x-2) \, dx$

 

[karush]

Re: 15.3 Express and integral for finding the area of region bounded by:

i was courious if this could be iterated into a double integral

 

[MarkFL]

Re: 15.3 Express and integral for finding the area of region bounded by:

i was courious if this could be iterated into a double integral

Yes. First, since we are simply finding the area $A$ bounded by $R$, the integrand will be 1. Using vertical strips, and the work already provided by skeeter, we could write:

$$A=\int_0^3\int_{x-2}^{2\sqrt{x}}\,dy\,dx+\int_3^{4+2\sqrt{3}}\int_{\frac{x}{3}}^{2\sqrt{x}}\,dy\,dx$$

Can you write the intergrals using horizontal strips?

 

[karush]

Re: 15.3 Express and integral for finding the area of region bounded by:

Yes. First, since we are simply finding the area $A$ bounded by $R$, the integrand will be 1. Using vertical strips, and the work already provided by skeeter, we could write:

$$A=\int_0^3\int_{x-2}^{2\sqrt{x}}\,dy\,dx+\int_3^{4+2\sqrt{3}}\int_{\frac{x}{3}}^{2\sqrt{x}}\,dy\,dx$$

Can you write the intergrals using horizontal strips?

but isn't this still 2 integral sets just added together ??

 

[MarkFL]

Re: 15.3 Express and integral for finding the area of region bounded by:

but isn't this still 2 integral sets just added together ??

Yes, but given the nature of the bounded region, there is no way I know of to express it as a single integral without a translation and conversion to polar coordinates.

We could translate everything down 1 unit and 3 units to the left:

View attachment 7265

Can you determine the limits in polar coordinates?