MHB 15.1.55 Find the mass of the plates with the following density functions

[karush]

$\textsf{A thin rectangular plate, represented by a region $R$ in the xy-plane}\\$
$\textsf{has a density given by the function p(x,y);}\\$
$\textsf{This function gives the area density in units such as $g/cm^2$}\\$
$\textsf{The mass of the plate is $\displaystyle\iint\limits_{R}p(x,y)dA$}\\$
$\textsf{Assume that $R=[(x,y): 0 \le x \le \frac{\pi}{2}, \, 0 \le y \le \pi]$}\\$
$\textsf{and find the mass of the plates with the following density functions}\\$

$\textit{a. $p(x,y)=10+\sin{x}$ (will do this one first)}\\$

\begin{align*}\displaystyle
M&=\int_0^\pi \int_0^{\frac{\pi}{2}} 10+\sin{x} \, dx\\
&=\int_0^\pi
\left[10x-\cos\left(x\right)\right]_0^{\frac{\pi}{2}} \, dy\\
&=\int_0^\pi 5{\pi}+1 \, dy\\
\end{align*}

hopefully so far ?

$\textit{b. $p(x,y)=10+\sin{y}$(this plotted outside the limits ?)}\\$
$\textit{c. $p(x,y)=10+\sin{x}\sin{y}$ (I tried to plot this in desmos but ?)}\\$

 

[Greg]

Re: 15.1.55 find the mass of the plates with the following density functions

Looks good except you forgot a $dy$ at the end of the right-hand side of the first line of your working for $M$.

 

[karush]

Re: 15.1.55 find the mass of the plates with the following density functions

Looks good except you forgot a $dy$ at the end of the right-hand side of the first line of your working for $M$.

ok

not sure what the slam dunk method is for
determining who to shrink first the x or the y

oh, started calc III class today (6 students) not sure who will still be alive by christmas

it was on parametric curves

 

[HallsofIvy]

Re: 15.1.55 find the mass of the plates with the following density functions

ok

not sure what the slam dunk method is for
determining who to shrink first the x or the y

oh, started calc III class today (6 students) not sure who will still be alive by christmas

it was on parametric curves

I really don't know what you mean by "shrinking" x or y. With a rectangular region, a\le x\le b and c\le y \le d you can integrate with respect to either variable first:
\int_a^b\int_c^d f(x, y)dy dx= \int_c^d\int_a^b f(x, y)dx dy.

Further these problems have nothing to do with 'parametric curves'.

 

[karush]

Re: 15.1.55 find the mass of the plates with the following density functions

I really don't know what you mean by "shrinking" x or y. With a rectangular region, a\le x\le b and c\le y \le d you can integrate with respect to either variable first:
\int_a^b\int_c^d f(x, y)dy dx= \int_c^d\int_a^b f(x, y)dx dy.

Further these problems have nothing to do with 'parametric curves'.

some of the homework questions were:
which Integral should be done first
to make it easier.

the class itself was on parametric curves
unrelated to the post

 

[karush]

Re: 15.1.55 find the mass of the plates with the following density functions

\begin{align*}\displaystyle
M_a&=\int_0^\pi \int_0^{\frac{\pi}{2}} 10+\sin{x} \, dx \, dy\\
&=\int_0^\pi
\left[10x-\cos\left(x\right)\right]_0^{\frac{\pi}{2}} \, dy \\
&=\int_0^\pi 5{\pi}+1 \, dy\\
&=\left(5{\pi}+1\right)y|_0^{\pi}\\
&=5\pi^2+\pi
\end{align*}

assume ok now for b and c

$\textit{b. $p(x,y)=10+\sin{y}$}\\$
$\textit{c. $p(x,y)=10+\sin{x}\sin{y}$}\\$

 

[karush]

Re: 15.1.55 find the mass of the plates with the following density functions

$\textit{b. $p(x,y)=10+\sin{y}$}\\$
\begin{align*}\displaystyle
M_b&=\int_0^\pi \int_0^{\pi/2} 10+\sin{y} \, dx \, dy\\
&=\int_0^{\pi} 10y-\cos(y)\Big\vert_0^{\pi/2} \, dy \\
&=\int_0^{\pi} 1-5\pi \, dy \\
&=(1-5\pi)y \Big\vert_0^{\pi}\\
&=\pi - 5\pi^2
\end{align*}

hopefully

 

[MarkFL]

Re: 15.1.55 find the mass of the plates with the following density functions

When you integrate w.r.t $x$, then you treat $\sin(y)$ as a constant...:)