All Questions
Tagged with bounds-of-integration multivariable-calculus
37
questions
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
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Triple Integral - Use symmetry for center of mass question?
I am unsure when to use symmetry with triple integrals.
Can I use symmetry for this centre of mass question?
$E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; ...
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Triple integral (mass) - setting up region between planes and parabolic cylinder
I am trying to set up the following triple integral using the xy plane.
$E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; \quad \rho(x, y, z)=8$.
I set up ...
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Double integral: technique to derive the limitations of $y$ (or $x$)
$$\iint_Cy dxdy, \quad C=\{(x,y)\colon0\leqslant x\leqslant2+y-y^2\}.$$
It is simple to see $x=2+y-y^2$ is a parabola with the symmetry axes is $x$ and the vertex $(9/4,1/2)$. It is easy to find the ...
- 7,792
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How does one calculate the area of a set?
The set is $M=\{(x,y)\in\mathbb{R}^2:|x|+|y|\leq 1\}$.
Question: How do you calculate the area of $M$? More specific, how do you find the bounds of integration?
Attempt: I tried to solve the ...
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Changing the Order of Integration in a Triple Integral
I'm currently studying for my multivariable calculus exam and I've come across a problem that I can't seem to solve. I have a triple integral with the order of integration $dz \, dy \, dx$ and I need ...
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Double integral of $1/(x^2+y^2)$ restricted to $x^2+y^2\leq2$ and $x\leq1$
Find $$\iint_D \frac{1}{(x^2+y^2)^2}dA$$ where $$D = \left\{ (x,y): x^2 + y^2 \leq 2 \right\} \cap \left\{ (x,y): x \geq 1 \right\}$$
Because of the prevalence of $x^2+y^2$ terms here, I figured we ...
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Integral bounds for $x\geq yz$
I am having trouble understanding the integral bounds.
From what side should my understanding go (first or second?):
first: as $z$ is between zero and one, $y$ is also between zero and one, thus $x$ ...
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Why does changing integral bounds get me the wrong answer?
Full disclaimer, this is a homework question.
While solving this question, I came upon the integral $$\int_{-r}^{r}\frac{b\tan^{-1}(\theta)}{2}\sqrt{r^2-x^2} dx$$ Proceeding with trig substitution I ...
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Help me understand easy (not for me) concepts in volume integral
Keep looking at the page for an hour.
Still not sure how I can get the sloping surface of $x+y+z=1$ and integration ranges for $x, y, z$. and why their range is different too.
The book keeps throwing ...
user808793
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Double integral setup - Uniform distribution
I am currently trying to understand a specific component of a probability problem involving setting up the proper bounds on a double integral.
In the problem, $X_1$ and $X_2$ are independent Uniform $(...
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$\iiint_M (x+y+z)\,dx\,dy\,dz$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$
$$\iiint_M (x+y+z)\,dx\,dy\,dz$$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$. How would I express this with the correct bounds? Once I have the bounds I can continue on my own but I need ...
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Determining Bounds of Theta for Double Integration of a Polar Region
Given the region $$ R:(x, y | x^2 + y^2 \le 4x) $$ And given the function $f(x, y) = \frac{x}{\sqrt{x^2+y^2}}$, find the double integral of the polar region.
So upon sketching the graph we get a ...
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Correct limit of integration when it difficult to visualize the region of integration?
I was running into a problem where I need to evaluate some probability over a region $D$.
A toy example would be this probability $\Pr \left[ {\underbrace {Y < \frac{{5\left( {X + 7} \right)}}{{9X}}...
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Is this the correct intersection between the two integration region?
My research require me to integrate this separable function $f\left( {x,y,z,t} \right) = f\left( x \right)f\left( y \right)f\left( z \right)f\left( t \right)$
over the region $D = {D_1} \cap {D_2}$
...