All Questions
Tagged with bounds-of-integration integration
72
questions
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Prove/disprove upper bound and lower bound of the Integral
Hey I need to Prove or disprove this sentence:
$$
\frac{4}{9}(e-1) \leq \int_0^1 \frac{e^x}{(1+x)(2-x)} \, dx \leq \frac{1}{2}(e-1)
$$
using the infimum and supremum method for integrals, where m and ...
- 1
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1
answer
46
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Calculating Volume of Spherical Cap using triple integral in cylindrical coordinates and spherical coordinates
Given a sphere above of $xy$-plane with center $(0,0,0)$ and radius $2$ (the equation $z=\sqrt{4-x^2-y^2}$). Plane $z=\sqrt{2}$ intersect the sphere.
I want calculate volume of spherical cap (orange ...
- 2,354
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1
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28
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Write triple integral as cylindrical coordinate of given region, confused in determining lower and upper bound.
Given $E$ is a region as follows:
$$E=\left\{0\leq x\leq 1, 0\leq y\leq \sqrt{1-x^2}, \sqrt{x^2+y^2}\leq z\leq \sqrt{2-x^2-y^2}\right\}.$$
Write triple integral
$$\iiint_\limits{E}xydzdydx$$
as triple ...
- 2,354
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1
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77
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
1
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0
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39
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How can I evaluate the bounds of this integral?
I have got this integral from a fourier transform:
$$\int_{-\infty}^0 e^{-ikx+x/4}+\int_0^{\infty} e^{-ikx-x/4}dx$$ Apparently the integrals give:
$$=\frac{1}{1/4 -ik}+\frac{1}{1/4 +ik}$$
But how? I'm ...
- 87
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75
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How is this property of definite integral derived?
The property:
$$
\int_a^b f(x) \, dx=\int_a^b f(a+b-x) \, dx
$$
Derivation given in my textbook:
Let $t = a+b-x$. Then $dt = -d x$. When $x=a, t=b$ and when $x=b, t=a$. Therefore,
$$
\begin{aligned}
\...
- 33
0
votes
1
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46
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For which lower bound of integration $a$ does a definite integral of $f(x)$ from $a$ to $x$ equal its antiderivative $F(x)$ with $C=0$?
For an arbitrary antidifferentiable function $f(x)$, my goal is to construct a definite integral of $f(x)$:
$$
\int_a^x f(t) dt
$$
which is equal to one of the infinitely-many antiderivatives of $f(x)$...
- 43
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Control $\int_0^\infty |\psi(x)|^2 dx$ by $\int_0^\infty \int_0^\infty K(x+y)\psi(x) \psi(y) dxdy$
Assume that $\psi(x)$ is bounded and integrable on $x \in [0,\infty)$ with $\int_0^\infty \psi(x) dx = 0$, and suppose that $K \colon (0,\infty) \to (0,\infty)$ is some kernel function satisfying $K(x)...
- 2,860
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27
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How to get a CDF value from a PDF when the required CDF is not within the defined area?
I have a density function f(x, y) = 1/2 for 0 ≤ x ≤ y ≤ 2 and 0 elsewhere. I am being asked to find the CDF value F(1, 3), but as you can see the three is past the range of the defined triangle, what ...
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1
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75
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triple integral pyramid bounds
I am still confused about how to set up bounds for double and triple integrals. My task is to set up bounds for a function that is a pyramid with edge coordinates $(5,+-5, 0)$, $(-5,+-5,0)$, $(0,0,4)$....
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106
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Changing the Order of Integration in a Triple Integral
I'm currently studying for my multivariable calculus exam and I've come across a problem that I can't seem to solve. I have a triple integral with the order of integration $dz \, dy \, dx$ and I need ...
- 11
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2
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119
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An upper bound for an integral
I saw many references using the following estimate but I couldn't prove it.
Given $T>0$ and $0 < b \leq \frac{1}{2}$, exist $C(b)$ constant that depends only on $b$ such that
\begin{equation}
\...
- 119
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2
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90
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Why don't the bounds in this definite integral change?
The question
This is probably a very basic question but I'm having a brain lapse and don't know why they didn't change the definite integral bounds from ($0 \rightarrow4$) to ($4 \rightarrow20$). I ...
- 5
3
votes
1
answer
46
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How do I find the bounds of this particular integral?
I want to convert this integral to Polar Coordinates to solve it: $\int_{0}^{2}\int_{0}^{\sqrt{y}}4xy^{2} \, dx \, dy$
What would be the bounds of $r$ and $\theta$ be?
I know how to solve the integral ...
0
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0
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54
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Show that the sum of two integrals is finite.
How to easy show that
\begin{equation} \int_0^1 \frac{1-e^{-x}}{x}dx+\int_1^M\frac{-e^{-x}}{x}dx \end{equation} is less than finite number?
- 31