Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
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Double Integrals
$(a)$ Sketch the region of integration in the integral
$$\int_{y=-2}^{2} \int_{x=0}^{\sqrt{4-y^2}} x e^{{(4-x^{2})}^{3/2}} dx dy$$
By changing the order of integration, or otherwise, evaluate the ...
- 403
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Keyhole integration
Ok so my lecturer gave us this powerful lemma for doing contour integrals over a semi-circle. The lemma is:
Let $C_{R}$ be the contour defined as $\{z \in \mathbb{C} \mid z = R e^{i \theta } , \...
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Integrate in Mathematica takes forever
I'm trying to calculate the length of a curve from a polynomial in Mathematica and for some reason Integrate doesn't complete, it just runs until I abort the execution.
The polynom:
...
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Riemann integral question
Suppose that $ f : [a,b] \rightarrow \mathbb{R}$ is Riemann integrable on $[a,b]$ and $g:[a,b] \rightarrow \mathbb{R}$ differs from $f$ at only one point $x_0 \in [a,b]$, that is, $g(x)=f(x)$ for $x \...
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Having such integral, how to optimize it in maple?
So we have :
(1/3)*sig0*h^3*(int(int(sin((1/3)*arctan(y, x)), x = 0 .. r), y = 0 .. 2*Pi))
Is it possible to optimise it? (in maple or any other way...)
How I ...
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Equality for the Gradient
We have that $f : \mathbb{R}^2 \mapsto \mathbb{R}, f \in C^2$ and $h= \nabla f = \left(\frac{\partial f}{\partial x_1 },\frac{\partial f}{\partial x_2 } \right)$, $x=(x_1,x_2)$.
Now the proposition ...
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Changing between Maxwell equations in differential and integral formats?
It takes me a long time to think about the equations even in one format and also to deduce things with Stokes. So how can you swap between the equations? I am looking more on the lines that suppose ...
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Domains of Integration -- the kernel trick and box-muller
I wonder if there is any deeper connection between two "tricks" from applied math, the kernel trick and the box-muller algorithm for generating draws from a random normal.
The kernel trick, used in ...
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problems on Lebesgue integral
1) Given a measure space, $f$ a non-negative measurable function and $A$ in the $\sigma$-algebra such that $\mu(A)=0$, prove that $\displaystyle\int_{A} f\;d\mu=0$ .
My try:
$$0\leq \int_{A}f\;d\mu=\...
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Not sure how to go about solving this integral
$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$
I've let $u=x^2+3$ but can't seem to get the right answer.
Really not sure what to do.
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How to do numerical integration of a function with values known at a given point set (finite and discrete) over an area bounded by discrete points?
Let $D$ be the area bounded by a series of points $(x_i,y_i)_{i=1}^{N}$.(The area need not to be convex and the points are supposed to go along the boundary curve.)
Let $f$ be a function defined on $...
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How to calculate $\int_0^{2\pi} \sqrt{1 - \sin^2 \theta}\;\mathrm d\theta$
How to calculate:
$$ \int_0^{2\pi} \sqrt{1 - \sin^2 \theta}\;\mathrm d\theta $$
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Derivative question
I need to find the derivative of:
$$
h(x) = \int_{0}^{x^2} (1-t^2)^{1/3} \, dt
$$
Would the answer to that just be:
$$
(1-x^4)^{1/3}?
$$
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Unbounded open set whose characteristic function is integrable in extended sense
Put simply, I would like to know if there is an unbounded open set (of $\mathbb{R}^n$) whose characteristic function is integrable in extended sense. I get the suspicion that something like the area ...
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Where does this 1 come from when balancing this integral equation?
$$ \int e^{ax}\cos(bx)\,\mathrm dx = \frac1{a}e^{ax}\cos(bx) + \frac{b}{a^2}e^{ax}\sin(bx) - \frac{b^2}{a^2}\int e^{ax}\cos(bx)\,\mathrm dx$$
$$\left(1 + \frac{b^2}{a^2}\right)\int e^{ax}\cos(bx)\,\...
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