All Questions
Tagged with integration multivariable-calculus
5,463
questions
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
I tried solving this problem as follows:
Equation of the cylinder $x^2+(y-a)^2=...
- 3,065
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votes
1
answer
61
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
I tried solving this problem as follows:
The equation ...
- 3,065
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1
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62
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere
This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$.
Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
- 5,450
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2
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79
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What does the integral $\int_0^V d\vec{x}$ mean/represent and how is it calculated?
My question is what does an integral such as
$$\int_0^V d\vec{x}_1$$
mean exactly?
Here is the context in which this type of integral arose.
I am following a thermodynamics course that has a section ...
- 5,021
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0
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29
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integration by parts $\int_{\mathbb{R}^N} -\Delta u (u-M)\varphi dx$
I'm trying to understand how it integrates by parts the following integral. This belong to the academic paper Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$ with DOI 10.4171/JEMS/217. ...
- 89
1
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1
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283
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Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
- 3,277
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20
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How to integrate function involving gradient of multivariate function?
Let's say we have a function f(x,y) = $x^2 + y^2$ , where x and y are real numbers and I have to compute the integral
$ I = \int g(v) \triangledown_{v} f(v) dv $, where v = [x, y]$^T$ and g(v) is a $R^...
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77
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
5
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1
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158
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Calculate the volume of intersection sphere and cone using triple integral
Compute the volume under sphere $x^2+y^2+z^2=4$ above $xy$-plane and above cone $z=\sqrt{x^2+y^2}$ using triple integral.
I try to plot as follows:
I try using triple integral
$$\int\limits_{-\sqrt{2}...
- 2,354
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1
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77
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Change the order of integration: $\int_0^2\int_y^{2y}xy dxdy$
I was presented on a class today the following integral and asked to calculate it and then try to change the order of integration: $$\int_0^2\int_y^{2y}xydxdy$$
As the integral is pretty simple I didn'...
- 11
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How to solve the integral $\iint_D(x^2+y^2)^{-2} dxdy$ with where $D$ is $x^2+y^2\leq2, x\geq 1$ and after that the same integral but with $x\leq1$. [closed]
So my main question is after I use the Jacobian how do I write the new space $D^*$ when I substitute $x=r\cos θ$ and $y=r\sin θ$. I know that I will find what the range of $θ$ is by simply finding the ...
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37
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How to solve the $\iint_D \frac{(x-y)^2}{1+x+y}dxdy$ where D is the trapezoid with edges $(1,0),(0,1),(2,0),(0,2)$ and using $u=1+x+y$ and $v=x-y$
I know that you need to find the Jacobian of $u,v$ and multiply it with the existing $f(x,y)$ inside the integral. But how can I then find the limits of integration for $u$ and $v$ after that since ...
3
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1
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64
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Find volume of body between surfaces
Problem: Find volume of body defined as follows:
$z^2=xy$, $(\frac{x^2}{2}+\frac{y^2}{3})^4=\frac{xy}{\sqrt{6}}$, $x, y, z \ge 0$.
My solution:
So we're working in the all positive octant of the ...
- 41
4
votes
3
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134
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Find area of figure and volume of body
Problem
Find area:
$$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2=\frac{xy}{c^2},\qquad a,b,c > 0$$
My solution:
The figure encloses area under itself so we're looking at: $$\left(\frac{x^2}{a^2}+...
- 65
2
votes
1
answer
45
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region above the cone and inside the sphere
I've come across a problem where
\begin{array}{c}
\mbox{I have to compute the volume of the region}
\\ \mbox{above the cone}\
z^2 = \sqrt{x^2 + y^2}\ \mbox{and inside the sphere}\ x^2 + y^2 + z^2 = 1
\...
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