# Questions tagged [integration]

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

66,758 questions
Filter by
Sorted by
Tagged with
5 views

### A question about the derivation of Poisson formula with respect to 2-dimensional wave equation

In section 2.4 (wave equation) of Partial Differential Equation (Evans), the author used the descent method to derive the Poisson's formula for two-dimensional wave equation. For the $n=3$ case, the ...
• 386
26 views

### What does $d^n\textbf{x}$ mean in this context?

I found the following on Wikipedia. Integration over more general domains is possible. The integral of a function $f$, with respect to volume, over an $n$-dimensional region $D$ of $\mathbb{R}^{n}$ ...
• 335
1 vote
62 views

### Showing $\int _{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$

Show that $$\int_{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$$ My method was this: I tried using $x \to \pi/4-x$ conversion but that doesn't lead to ...
48 views

1 vote
27 views

### Computing areas using Green's theorem

I want to compute the area of the surface $B$ with boundary parametrised by $$\gamma(t)=\left(\begin{array}{c} \sin t \\ 4 \cos ^{2} t+\cos t \end{array}\right), \quad t \in[0,2 \pi]$$ ...
• 13
27 views

• 792
40 views

48 views

• 2,817
36 views

### can the upper integral of a function be larger than its domain? [closed]

When looking at Darboux integrals, is it possible for the upper integral to be larger than the product of the length closed interval and the range?
• 1
32 views

### Integrating $3 \cos^2 x \sin x$ [closed]

$$I = \int_0^1 3 \cos^2 x \sin x \,{\rm d}x$$ Please show it step by step and explain each step.
55 views

### If $S$ is set of lower sums for partitions having $n$-equal subintervals, does $\sup S$ equal the $\sup$ of the set of lower sums for all partitions.

Firstly, here are two relevant definitions: Let $a \lt b$. A partition of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$. Suppose $f$ ...
• 4,307
1 vote
31 views

### Log-normal distribution: Why is this a density function?

I want to prove that $$f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp(-\frac{\ln(x)^2}{2\sigma^2})$$ for $x>0$ and $f(x)=0$ for $x\leq 0$ is a density function, with $\sigma > 0$. So what I have to ...
• 449
22 views

### Trouble understanding a criteria for integrability

Let $f : [0, 1] → \mathbb{R}$ be bounded and $f ≥ 0$. Prove that if the set $\{x\in[0,1]|f(x)\geq \lambda \}$ is finite for every $λ > 0$ then $f$ is integrable. I'm having trouble understanding ...
• 488
1 vote
24 views

### Integrating distributions over submanifolds

Distributions may be "integrated against" (i.e. evaluated on) test functions, the following notation (for $T$ a distribution) is fairly common: $$T(f) = \int T(x) f(x)\, \mathrm{d}x$$ I'm ...
• 1,854
13 views

### Relation between normalized product of integrals of two functions and integral of product of two normalized functions

Could someone let me know if there is any relation between A and B, if $A=\frac{\int^L_0 f(x)g(x)dx}{L}$ and $B=\frac{\int^L_0 f(x)dx\cdot\int^L_0 g(x)dx}{L^2}$
30 views

• 24k