All Questions
Tagged with integration solution-verification
1,327
questions
-1
votes
2
answers
59
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$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$
Calculate:
$$\int_A xyz \ d \lambda_3$$
Solution:
We know that: $x^2 + y^2 + z^2 > 0$ and therefore $2x + 2y > 0 \iff x + y > 0$
...
- 1,086
1
vote
2
answers
128
views
Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$
Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$.
I have tried a couple methods. As a first method, it is tempting to bound the integrand above ...
- 2,782
1
vote
1
answer
39
views
What is the fault in this method of finding second moment of area of a circle
I am trying to find the second moment of area of a circle about a diameter using first principles.
Place the centre of the circle at the origin of XY-plane. Now consider a tiny circular sector with an ...
- 141
0
votes
1
answer
35
views
Showing integrability of f+g and additivity of the Darboux integral
I am currently working on the following question from Measure, Integration & Real Analysis by Sheldon Axler:
Suppose $f,g:[a,b]\to\mathbb{R}$ are Riemann (Darboux) integrable on $[a,b]$. Prove ...
- 508
5
votes
0
answers
62
views
Integrating the Beta Function
As a learning exercise, I am trying to find the mean and variance of the Beta Probability Distribution (https://en.wikipedia.org/wiki/Beta_distribution) from first principles (i.e. Method Of Moments):
...
- 399
4
votes
1
answer
69
views
Determine whether $\{\int_0^1 f^2(t)dt = 0\}$ is a subspace of $C([0,1],\mathbb{C})$, complex-valued case
I am currently self-studying through a text on linear algebra, and one of the problems asks to determine whether the set $\mathcal{U} = \{f \in C([0, 1],\mathbb{C}) \colon \int_0^1 f^2(t)dt = 0\}$ is ...
- 53
0
votes
1
answer
65
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Theorem 7.35 in Apostol's MATHEMATICAL ANALYSIS, 2nd edition: Is the continuity of $\alpha$ on $[a, b]$ essential in the hypothesis?
Here is Theorem 7.35, in Chapter 7, in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition:
Assume $f \in R$ on $[a, b]$. Let $\alpha$ be a function ...
- 26.9k
1
vote
1
answer
42
views
Evaluate $\iint_S\vec{F}$ where $S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above and below by two planes.
Use the divergence theorem to evaluate $$\iint_S\vec{F}$$ where $\vec F(x,y,z)=\langle xy^2,x^2y,y \rangle$ and $\vec S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above ...
- 3,065
7
votes
6
answers
691
views
The method of substitution in the problem of finding the integral
My teacher gave me a simple problem:
Find $\int \dfrac{x}{\sqrt{x+1}} \, dx$.
This is how I approached it:
I set $u = \sqrt{x+1}$, which implies $u^2 = x + 1$, thus $2u \, du = dx$.
Therefore,
$$
\int ...
- 817
0
votes
2
answers
110
views
How to integrate $\int\frac{x(1-x)}{(1+x)}dx$
Q)How to integrate $$\int\frac{x(1-x)}{(1+x)}dx$$
My Approach :
I know how to integrate these types of integrals. We have to substitute $(1+x)=u$. Then $dx=du$.
Therefore the integral will become :
$$\...
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
2
votes
2
answers
131
views
Solving the Stacked Integral $\int_1^{\int_1^{\int_1^{\int_1^{\dots}}2xdx}2xdx}2xdx$
A friend sent me the integral:$$\int_1^{\int_1^{\int_1^{\int_1^{\dots}}2xdx}2xdx}2xdx$$
I tried my best with the formatting, but if it's not clear, it is the integral of $2x$ with a lower bound of $1$...
- 741
0
votes
1
answer
30
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Under what hypotheses is the primitive function bijective?
I am trying to determine under what assumptions the function
$$F:(0,\infty) \to (0,\infty),$$
defined by
$$F(t) = \int_{0}^{t} f(s)ds$$
is a bijection. For injectivity, simply require that $f$ ...
- 3,004
0
votes
0
answers
46
views
Solution of Cauchy problem by Kirchhoff’s formula
while studying the Cauchy problem
\begin{array}{l}
{u_{tt}} - {\nabla ^2}u = 0,x \in {\mathbb{R}^3}\\
u\left( {x,0} \right) = 0\\
{u_t}\left( {x,0} \right) = f\left( {x} \right)
\end{array}
the ...
0
votes
0
answers
34
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Why is this proof about integration correct? [duplicate]
I have already asked about this particular integral, but I am not sure if this reasoning makes sense.
From the equality
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-x^2-y^2}\ dxdy=\int_{-\infty}...
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