All Questions
76
questions
4
votes
3
answers
134
views
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: $$\left(\frac{x^2}{a^2}+...
- 65
2
votes
1
answer
65
views
Can use one-variable integration to tell whether $R=\{(x,y):\;0\leq y\leq\left|\mathrm{sin}\frac{1}{x}|,\;0<x\ <1\right\}$ is rectifiable(has area)?
$R=\left\{(x,y):\;0\leq y\leq\left|\mathrm{sin}\frac{1}{x}\right|,\;0<x\ <1\right\}$, to determine whether it's rectifiable, can I use the method of one-variable integration to prove? Like this ...
- 123
0
votes
0
answers
38
views
Calculating the area of a paraboloid inside a cylinder
So I have to calculate the area of a paraboloid $2z=\frac{x^2}{a}+\frac{y^2}{b}$ inside a cylinder $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. So I plugged in my formula:
$$\iint_R\sqrt{1+\left(\frac{\...
- 489
2
votes
1
answer
77
views
Area Calculation of Region A (Using Double Integral)
I'm calculating the area of the region $A$ defined by the following constraints:
$$ A = \{(x, y) \mid x^2 + y^2 \geq 2, \; x^2 + y^2 \leq 2x, \; y \geq 0\} $$
To calculate the area of the region $A$, ...
- 698
0
votes
0
answers
47
views
Formulating the Integral for Calculating the Mass of a Plate with Given Density
I am seeking guidance on formulating the integral to calculate the mass of a plate with density $d(x, y) = x + y$, whose shape is defined by the following figure:
I made the next:
The mass $M$ of the ...
- 698
3
votes
2
answers
126
views
Area of $x^{10}+y^{10}\leq 1$
Whilst looking at someone's vector calculus problem, they mentioned that, making use of Green's Theorem, they had to express the line integral of the boundary of $x^{10}+y^{10}\leq 1$ in terms of its ...
- 2,613
0
votes
0
answers
24
views
inferring sign of a double integral over a general region D from double integrals of all boxes that lie inside D and containing D
Suppose a double integral
$$
I_{D}:=\iint\limits_D f(x,y) dx dy
$$
over a specific type of region $D=\{(x,y) : x_{\min}\leq x \leq x_{\max}, h_1(x) \leq y \leq h_2(x)\}$. Note that $h_1, h_2$ are ...
- 71
1
vote
2
answers
61
views
Region area with double integral
I need to find the area $\{(x, y) \in \mathbb{R}^2 : 0 < 2x < y < 3-x^2\}$ as the double integral $\int(\int dx) dy$.
I came up with:
$$\int_0^2\int_0^{\frac{1}{2}y}dxdy + \int_2^3\int_0^\...
- 653
1
vote
1
answer
35
views
Extra piece in this double integral
I'm asking for help in understanding how to end this exercise. I will write down my procedure, which is incomplete.
Calculate the area of the region defined by the circumference of centre $C = (1,2)$ ...
- 3,482
1
vote
1
answer
45
views
Interpretation of the double integral over a region
I am studying multivariable calculus, and I came up with a doub that I cannot solve, or I believe I udnerstood wrong.
Say I have to calculate the following integral:
$$\int_T (x^2 + y)\ \text{d}x\text{...
- 3,482
1
vote
1
answer
71
views
Determine area bounded by $(x+y)^4 = ax^2y$
Determine the area of region $U$ bounded by the graphic of the curve:
$$(x+y)^4 = ax^2y,\ a > 0 \quad \text{(loop in the first quadrant)}$$
I used polar coordinates and I arrived to:
$$\iint r\,dr\,...
- 89
7
votes
2
answers
109
views
What do these double integrals represent?
Question
I have three double integrals:
(A) $$\int_{0}^{1}\int_{x^2}^1 dydx$$
(B) $$\int_{0}^{1}\int_{x}^{2x} x^2 dydx$$
(C) $$\int_{0}^{1}\int_{-y}^{y} dxdy$$
These need to be matched to the ...
- 1,900
0
votes
1
answer
301
views
Variable substitution for double integral
I'm calculating the following integral:
$$I=\iint_{D}2cos((x+y)^2)\, dxdy,$$
where $D=\left \{ (x,y):1 \leq x+y \leq 3, x \geq 0, y \geq 0 \right \}$.
To get around the fact that I would have to ...
2
votes
0
answers
72
views
Did I construct my integral correctly? - Find the surface area of the cone $z = \sqrt{x^2 +y^2}$ bounded in the cylinder $x^2 + y^2 = 4x$
In a practice test, there's a question that wants us to find the surface area of a cone $z = \sqrt{x^2 +y^2}$ that's bounded by $x^2 + y^2 = 4x$.
I'm really shaky on the subject matter and since ...
- 507
5
votes
1
answer
289
views
How to calculate or estimate the area of such implicit region about function $x^{\frac{1}{x+\frac{1}{x}}}$
How to estimate or calculate the area enclosed by the implicit equation.
$$x^{\frac{1}{x+\frac{1}{x}}}+y^{\frac{1}{y+\frac{1}{y}}}=\mathrm{e}$$
It is possible to prove that the area is between $13$ ...
- 321