Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
138,148 questions
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Changing the order of integration in an iterated integral with a single varible function
I am trying to consider a double integral:
$$
\int_t^\infty \int_s^\infty f(r) dr ds <+\infty
$$
where $f:\mathbb{R} \to \mathbb{R}$ is a smooth function, but NOT a non-negaitive function. And the ...
- 27
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Proof of Euler identity using ODE
I am writing an article to prove Euler identity :$e^{i\pi}+1=0$
Here the main part:
Consider the function :$ \mathbb{R} \rightarrow \mathbb{C}, f(x)=e^{ix}$
Differentiating twice,we get : $f''(x)=-f(x)...
- 58
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
...
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Base of the line integration [closed]
Does line integral integrate over the projection of a 3d curve onto the x-y plane or over the 3d curve itself as the base of the integration?
Thanks
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Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$
My teacher used integration by parts to solve the problem like so:
$$\int_0^2 xd(\{x\})
=[x\{x\}]_0^2-\int_0^2 \{x\}dx\\
=0-\int_0^1 xdx-\int_1^2 (x-1)dx$$
which comes out to -1. But when I was ...
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How to integrate $\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$ analytically?
I'm trying to solve the integral
$$\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$$
I do know that a similar integral
$$\int \frac{12x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{...
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True or false? If $\lim n|x_n-x_{n+1}|=0$ then $\{x_n\}$ converges. [duplicate]
I want to know whether it is true that if a real sequence $\{x_n\}_{n=1}^\infty$ satisfies $\lim\limits_{n\to\infty} n|x_n-x_{n+1}|=0$ then it converges.
I guess it is false but I can't find a ...
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Can I treat the limit as a constant? [closed]
Imagine that the limit as $h$ approaches infinity of $f(1 + hx)$ is $g(x)$.
$$\lim_{h\to \infty} f(1 + hx) = g(x)$$
Can I then say that $f(x)$ is equal to the limit as $h$ approaches infinity of $g\...
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Is my understanding correct about the inverse relationship between derivatives and integrals?
I've learned that derivatives and integrals are inverse operators, but am not completely sure why. I've looked at many resources to understand why, and here goes.
The integral gives a sum of the total ...
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Find out the distance between centers of two intersecting semi-ellipses $x^2/a^2+y^2/b^2=1$.
There are two identical semi-ellipses, one with center at the origin $O$, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, and the other at $R$, $\frac{(x-d)^2}{a^2}+\frac{y^2}{b^2}=1$.
Find out the distance $d$ ...
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How to setup bounds for triple integrals [closed]
Set up (do not evaluate) triple integrals in spherical coordinates in the orders dρdϕdθ and
dϕdρdθ to find the volume of the cube cut from the first octant by the planes x = 1, y = 1
and z = 1.
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Dummy variable rule for indefinite integrals?
My friend is tutoring high school mathematics, and one of the techniques taught is to let an integral be $I$ then get $I = abc - I$ so that $I = abc/2.$ For example, $$
I := \int e^x\cos{x} dx = eˣ \...
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Divergence Test [closed]
The divergence test is inconclusive for $f(x) = 1/x$. The sum of the $1/n$'s is in fact divergent, $n\in \mathbb{N}$ and $n$ in $[1, ∞)$. We can say the same for the function $g(x) = 1/x^p$, with $0 &...
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Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
Given, $$f(x) = x^3 - 3x + 1$$
I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$.
By analyzing the graph of $f(x)$, we can observe the local ...
- 964
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Showing $\int_0^a\left(f(x)-\frac12x\right)^2dx\leq\frac1{12}a^3$ for $f(x)\geq0$ satisfying $\left(\int_0^tf(x)dx\right)^2\geq\int_0^tf^3(x)dx$ [duplicate]
Problem:
Given positive value $a$, we have $f(x) \geq 0,\forall x\in[0, a]$, and
$$\left(\int_0^t f(x) dx\right)^2 \geq \int_0^t f^3(x)dx, \quad\forall t \in [0, a]$$
Show that
$$\int_0^a \left(f(x)-\...
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