All Questions
330
questions
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61
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A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.
Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$
Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
- 3,065
1
vote
1
answer
69
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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
3
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
5
votes
1
answer
158
views
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
$$\int\limits_{-\sqrt{2}...
- 2,354
3
votes
1
answer
64
views
Find volume of body between surfaces
Problem: Find volume of body defined as follows:
$z^2=xy$, $(\frac{x^2}{2}+\frac{y^2}{3})^4=\frac{xy}{\sqrt{6}}$, $x, y, z \ge 0$.
My solution:
So we're working in the all positive octant of the ...
- 41
0
votes
0
answers
45
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Calculating the volume of a body given by $g(x,y,z) = xyz$ using triple integrals
I need help calculating the volume of the region/body/solid given by
$D = \{(x,y,z) \colon 0 \leq xyz \leq 8 \text{ and } 0 \leq x \leq 2, 0 \leq y \leq 2, 0 \leq z \leq 2\}$,
I am supposed to do it ...
- 31
2
votes
3
answers
95
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How do I express the region bounded by $z=x^2, z+y=1, z-y=1$ as a z-simple triple integral?
I've been able to figure out the y-simple and x-simple triple integrals, but I'm having trouble with the z-simple one. I split the region in half over the x-axis, from what I can see, the projection ...
- 43
1
vote
1
answer
80
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Evaluating triple integral under linear transformation
Given linear transformation: $$T: \mathbb R^3 \to \mathbb R^3, T(x,y,x)= (3y+4z, 2x-3z, x+3y) $$
We need to evaluate the triple integral :
$$\int \int \int_ {T(C)} (2x+y-2z) dx dy dz $$
where $C= \{{(...
- 1,239
3
votes
2
answers
306
views
Calculating Volume of "fat sphere" defined by $x^8+y^8+z^8 = 64$
A young friend of mine had a homework problem involving a surface integral over the surface of the equation $x^8+y^8+z^8=64$. The homework problem was solvable using the divergence theorem, but he ...
- 233
0
votes
0
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137
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$\int_D 1$, where $D:=\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = r^2, (r, z) \in C\}$. (Problem 3-29 in "Calculus on Manifolds" by Michael Spivak)
The following problem is a problem in the section "FUBINI'S THEOREM".
Problem 3-29.
Use Fubini's theorem to derive an expression for the volume of a set of $\mathbb{R}^3$ obtained by ...
- 1,138
2
votes
1
answer
34
views
Evaluate the volume between a parabola and a sphere in high dimension
Consider the unit ball $\mathbb{B}^{n+1}:=\{\boldsymbol{x}\in\mathbb{R}^{n+1}:\|\boldsymbol{x}\|_2\le 1\}$ and a parabola $p:\mathbb{R}^{n}\rightarrow\mathbb{R}$ for which $p(\boldsymbol{x}):=\rho\|\...
- 163
3
votes
2
answers
429
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Finding the Volume of Region in Three-Dimensional Space.
I am currently working on calculating the volume of the region $W$ in $\mathbb{R}^3$, defined by the constraints:
$$W=\{ (x,y,z)\in\mathbb{R}^3 ∣ z≥2, x^2+y^2≤6−z, y≥x \} .$$
My approach involves ...
- 698
1
vote
1
answer
108
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Determine the mass of the solid, provided its density at a point $P(x,y,z)$ is proportional to the distance from the $XY$ plane.
I am trying to solve the following problem*
A solid can be described in three-dimensional space by the following system of inequalities:**
$$\begin{cases}
x^2+y^2+z^2\leq1\\
z\geq\sqrt{x^2+y^2}
\end{...
- 1,191
2
votes
1
answer
62
views
Volume of a solid bounded by surfaces [closed]
can someone help me with setting up the required integral?
The solid is bounded by $z = 4x^2 + 4y^2 -27$ and by $z = x^2+y^2$. Should I project this on $x-y$ plane or use cylindrical or spherical ...
- 631
1
vote
1
answer
139
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Calculate the volume of the body defined by $x^2+y^2+z^2≤1$ for which $x^2+z^2≥z$
Calculate the volume of the body defined by $x^2+y^2+z^2≤1$ for which $x^2+z^2≥z$
Attempt:
For the volume, I used this formula: $V=\iint y(x,z)dxdz$
I find it easier to express the problem through $y$ ...
- 53