Volume with double integral
$begingroup$
Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$.
From this, I got that the volume would simply consist of the region under $z = 1-x-(5/6)y$. And as $z=0$, the plane intersects the $x-y$ plane at $6x+5y=6$. Therefore, I thought the region was bounded by $y = x$, $6x+5y = 6$ and $x=0$. After rearranging the equations and drawing the diagrams, I got the following integral:
$$int _0^{ frac{6}{5}}int _x^{frac{6}{5}-frac{6}{5}x}1-x-frac{5}{6}y:dydx$$
This integral gave me a volume of $186/625$, but this was not correct.
Any help would be highly appreciated!
calculus integration multivariable-calculus volume
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
$endgroup$
add a comment |
$begingroup$
Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$.
From this, I got that the volume would simply consist of the region under $z = 1-x-(5/6)y$. And as $z=0$, the plane intersects the $x-y$ plane at $6x+5y=6$. Therefore, I thought the region was bounded by $y = x$, $6x+5y = 6$ and $x=0$. After rearranging the equations and drawing the diagrams, I got the following integral:
$$int _0^{ frac{6}{5}}int _x^{frac{6}{5}-frac{6}{5}x}1-x-frac{5}{6}y:dydx$$
This integral gave me a volume of $186/625$, but this was not correct.
Any help would be highly appreciated!
calculus integration multivariable-calculus volume
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
$endgroup$
$begingroup$
What is the correct answer?
$endgroup$
– K Split X
Nov 29 '18 at 2:29
add a comment |
$begingroup$
Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$.
From this, I got that the volume would simply consist of the region under $z = 1-x-(5/6)y$. And as $z=0$, the plane intersects the $x-y$ plane at $6x+5y=6$. Therefore, I thought the region was bounded by $y = x$, $6x+5y = 6$ and $x=0$. After rearranging the equations and drawing the diagrams, I got the following integral:
$$int _0^{ frac{6}{5}}int _x^{frac{6}{5}-frac{6}{5}x}1-x-frac{5}{6}y:dydx$$
This integral gave me a volume of $186/625$, but this was not correct.
Any help would be highly appreciated!
calculus integration multivariable-calculus volume
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
$endgroup$
Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$.
From this, I got that the volume would simply consist of the region under $z = 1-x-(5/6)y$. And as $z=0$, the plane intersects the $x-y$ plane at $6x+5y=6$. Therefore, I thought the region was bounded by $y = x$, $6x+5y = 6$ and $x=0$. After rearranging the equations and drawing the diagrams, I got the following integral:
$$int _0^{ frac{6}{5}}int _x^{frac{6}{5}-frac{6}{5}x}1-x-frac{5}{6}y:dydx$$
This integral gave me a volume of $186/625$, but this was not correct.
Any help would be highly appreciated!
calculus integration multivariable-calculus volume
calculus integration multivariable-calculus volume
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
edited Nov 29 '18 at 4:23
Key Flex
7,93461233
7,93461233
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
asked Nov 29 '18 at 2:24
sktsasussktsasus
1,010416
1,010416
$begingroup$
What is the correct answer?
$endgroup$
– K Split X
Nov 29 '18 at 2:29
add a comment |
$begingroup$
What is the correct answer?
$endgroup$
– K Split X
Nov 29 '18 at 2:29
$begingroup$
What is the correct answer?
$endgroup$
– K Split X
Nov 29 '18 at 2:29
$begingroup$
What is the correct answer?
$endgroup$
– K Split X
Nov 29 '18 at 2:29
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$
The bounds for $z$ is from $0$ to $dfrac{6-6x-5y}{6}$
The solid region onto the $xy$ plane is
$y=x, x=0, 6x+5y=6$
The bounds for $y$ is from $x$ to $dfrac{6-6x}{5}$
The bounds for $x$ is from $0$ to $dfrac{6}{11}$
The Volume is $$int_{0}^{frac{6}{11}}int_{x}^{frac{6-6x}{5}}dfrac{6-6x-5y}{6} dy dx=dfrac{6}{55}$$
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$
The bounds for $z$ is from $0$ to $dfrac{6-6x-5y}{6}$
The solid region onto the $xy$ plane is
$y=x, x=0, 6x+5y=6$
The bounds for $y$ is from $x$ to $dfrac{6-6x}{5}$
The bounds for $x$ is from $0$ to $dfrac{6}{11}$
The Volume is $$int_{0}^{frac{6}{11}}int_{x}^{frac{6-6x}{5}}dfrac{6-6x-5y}{6} dy dx=dfrac{6}{55}$$
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
$endgroup$
add a comment |
$begingroup$
$6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$
The bounds for $z$ is from $0$ to $dfrac{6-6x-5y}{6}$
The solid region onto the $xy$ plane is
$y=x, x=0, 6x+5y=6$
The bounds for $y$ is from $x$ to $dfrac{6-6x}{5}$
The bounds for $x$ is from $0$ to $dfrac{6}{11}$
The Volume is $$int_{0}^{frac{6}{11}}int_{x}^{frac{6-6x}{5}}dfrac{6-6x-5y}{6} dy dx=dfrac{6}{55}$$
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
$endgroup$
add a comment |
$begingroup$
$6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$
The bounds for $z$ is from $0$ to $dfrac{6-6x-5y}{6}$
The solid region onto the $xy$ plane is
$y=x, x=0, 6x+5y=6$
The bounds for $y$ is from $x$ to $dfrac{6-6x}{5}$
The bounds for $x$ is from $0$ to $dfrac{6}{11}$
The Volume is $$int_{0}^{frac{6}{11}}int_{x}^{frac{6-6x}{5}}dfrac{6-6x-5y}{6} dy dx=dfrac{6}{55}$$
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
$endgroup$
$6x+5y+6z = 6$, $y=x$, $x=0$ and $z=0$
The bounds for $z$ is from $0$ to $dfrac{6-6x-5y}{6}$
The solid region onto the $xy$ plane is
$y=x, x=0, 6x+5y=6$
The bounds for $y$ is from $x$ to $dfrac{6-6x}{5}$
The bounds for $x$ is from $0$ to $dfrac{6}{11}$
The Volume is $$int_{0}^{frac{6}{11}}int_{x}^{frac{6-6x}{5}}dfrac{6-6x-5y}{6} dy dx=dfrac{6}{55}$$
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
answered Nov 29 '18 at 2:29
Key FlexKey Flex
7,93461233
7,93461233
add a comment |
add a comment |
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$begingroup$
What is the correct answer?
$endgroup$
– K Split X
Nov 29 '18 at 2:29