All Questions
58
questions
2
votes
1
answer
66
views
What coordinate substitution should I perform to evaluate this triple integral?
I am trying to evaluate the following triple integral:
\begin{equation}
\int_{-1}^1 \int_{-\sqrt{4-4x^2}}^{\sqrt{4-4x^2}} \int_{\sqrt{4x^2 + z^2}}^2 ye^{4x^2 + y^2 + z^2} \, dy\, dz\, dx
\end{equation}...
4
votes
0
answers
53
views
evaluate the volume of solid
Consider the paraboloid $(\mathcal{P}): z=x^2+y^2$ and the plane $(\mathcal{Q}): 2x+2y+z=2$.
Let $\mathcal{S}$ be the solid region bounded above by $(\mathcal{Q})$ and below by $(\mathcal{P})$. Find ...
- 319
0
votes
2
answers
39
views
set the limits of integration of the spherical coordinates between two paraboloids and a plane
Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the ...
- 319
1
vote
1
answer
64
views
Area of a plane enclosed by a cylinder
I have the cylinder $x^2+y^2=R^2$ and the plane $z=ax+b$ ($R,a,b$ are constants). I need to find the area of the ellipse that is the region of the plane enclosed by the cylinder.
I've done it with a ...
- 45
0
votes
0
answers
26
views
How to approach multivariable integration problems?
I have a problem that seems to be a cylindrical conversion problem, but I could not find bounds for r.
The problem asked me to find the volume bounded by $z = (2x)^2 + y^2$ and $z+y^2 = 2$.
I first ...
3
votes
0
answers
67
views
Struggling with a Calculus III problem :$\iint x+y\ dS,$ where $S(u,v)=2\cos(u)\vec i+2\sin(u)\vec j+v \vec k$ and $0\le u\le \pi/2;\;0 \le v \le9.$
So I have had Calc III many moons ago but I cannot seem to solve this problem for my son who is taking it now. Worked it four ways and got four different answers. Hoping someone here can set me ...
- 31
1
vote
1
answer
58
views
Understanding the Meaning of $y \geq x$ in Cylindrical Coordinates during a Variable Transformation.
I am seeking clarity / intuition on the meaning of the condition $y \geq x$ when performing a change of variables from Cartesian to Cylindrical Coordinates for volume integration.
In the context of ...
- 698
4
votes
1
answer
187
views
Solving triple integral with cylindrical coordinates
We are told to evaluate the triple integral:
$$\iiint_E z dV$$
where $E$ is bounded by $x=4y^2+4z^2$ and $x=4$.
My attempt:
First I noticed that this represents a paraboloid on the x axis so I thought ...
- 151
-3
votes
1
answer
322
views
Find the volume of the solid bounded by $z=\sqrt[4]{x^2+y^2}$, $z=1$ and $z=\sqrt{2}$
Find the volume of the solid bounded by $z=\sqrt[4]{x^2+y^2}$, $z=1$ and $z=\sqrt{2}$. I have to calculate this using triple integral but the I'm stuck on finding the intervals.
6
votes
1
answer
147
views
Using change of coordinates to find the exact value of an integral
Use an appropriate change of coordinates to find the exact value of the integral
$$\int_{-\sqrt{3}}^{\sqrt{3}}\int_{-\sqrt{3-x^2}}^{\sqrt{3-x^2}}\int_{-3+x^2+y^2}^{3-x^2-y^2}x^2dzdydx$$
My work so far:...
- 1,900
0
votes
0
answers
52
views
How can I tell which function of two variables is larger?
In this case, $z = 1$ and $z = \sqrt{x^2 + y^2}$. How can I tell which function is bigger to choose the upper and lower bound?
2
votes
3
answers
215
views
Finding the volume with triple integrals
I want to find the volume of a function described by:
$$ G= \{(x,y,z)|\sqrt{x^2+y^2} \le z \le 1, (x-1)^2+y^2 \le 1\}$$
This question can be best solved in cylindrical coordinates. So if I follow that ...
0
votes
1
answer
212
views
Rewriting triple integrals rectangular, cylindrical, and spherical coordinates
Write three integrals, one in Cartesian/rectangular, one in cylindrical, and one in spherical coordinates, that calculate the average of the function $f(x, y, z) = x^2 + y^2$ on the region $E$ in the ...
2
votes
1
answer
68
views
Does order of integration matter, while integrating over a cone?
Suppose I want to find the center of mass of a cone, from its vertex. It is a right circular solid cone of radius $a$.
About the $z$ axis, the center of mass is given by : $$z_{com}=\frac{\int dm\,z}{\...
4
votes
1
answer
430
views
Volume inside an ellipsoid and offset cylinder
I want to find the volume of the region inside the ellipsoid $$\frac{x^2}{4}+\frac{y^2}{4}+z^2=1$$ and the cylinder $$x^2+(y-1)^2=1$$
I tried shifting the axes so that the cylinder was centered at ...
- 53