All Questions
Tagged with applications integration
63
questions
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How to solve an ODE where the rate is directly proportional to two amounts?
Two chemicals in solution react together to form a compound: one unit of compound is formed from $a$ units of chemical $A$ and $b$ units of chemical $B$, with $a + b = 1$. Assume the concentration ...
- 4,450
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Using the trapezoidal rule for the Maxwell-Boltzman function
Background
I approached my physics professor with question 1 from this LibreTexts resource. (at the bottom of the page), to better understand the material via self-study.
Question
Using the Maxwell-...
- 1,426
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What is the equation and area under curve for Covid load dynamics?
Covid virions on infection, replicate exponentially and once the body's defense system starts attacking it then it also seems to decrease exponentially.
Source
The time period when the PCR test is ...
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Sequence of Logic in Diffusion Problem DQ
Problem: If a tank is filled with 100 gallons of water and mistakenly added 300 pounds of salt. To fix the mistake the brine is drained at 3 gallons per minute and replaced with water at the same rate....
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Arc length vs Surface of revolution.
I don't understand why these two problems are solved differently here the first one $fig(1)$ and 2nd one $fig(2)$. Why did we take the limit $\displaystyle \lim_{r\to0^+}\int_r^\pi \sqrt{2-2cost}\...
user1062191
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63
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Why can we apply the surface area of revolution theorem to a spiral?
To find the surface area generated by revolving function f which is smooth on the interval [a,b] and $f(y) \ge0$ around the y-axis we can use the formula $$S=\int_a^b 2\pi rdl =\int_a^b 2\pi f(y)\...
- 318
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2
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When can I apply the trapezoidal rule?
An artificial lake has the shape illustrated below , with adjacent measurements 20 feet apart. Use suitable numerical method to estimate the surface area of the lake.
I know how to solve this problem ...
- 318
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Why can we say here that $\Delta x_i=dx$ as $i$ approaches infinity?
In the proof of the arc length formula we assume that an element of the arc length is $$\Delta L_i=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2}=\sqrt{1+\left(\frac{\Delta y_i}{\Delta x_i}\right)^2}\space \...
user1062191
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1
answer
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Calculus application question
My attempt:
Step 1: Find $x$ in terms of $t$.
$\frac{dt}{dx} = \frac{1}{-0.15x}$
$t = \frac{1}{-0.15}\ln(x) = x^{-1}(t)$
$x(t) = e^{-0.15t}+c$
However, here is where I am stuck. Without any extra ...
- 1,190
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311
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Using integration to find the population $x$ after a time $t$ years. Having a problem with getting a negative log input.
I'm a little bit confused by a question I came across. It says:
If there were no emigration the population $x$ of a county would increase at a rate of $2.5 \%$ per annum.
By emigration a county loses ...
- 71
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I've never been so confused (Application of Integral Calculus)
Here's a problem on Application of Integral calculus to find the work done in moving a particle. I was able to 'reach' the 'right answer'. But I'm totally confused and utterly dissatisfied with the ...
- 1,156
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2
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2k
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Work on a chain (applications of the integral)
A 10-foot-long chain weighs 25 lbs. And hangs from a ceiling. Calculate the work done in raising the lower end of the chain to the ceiling so that it is at the same level as the upper end.
Please, ...
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100
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Calculus applications - oil leaking from a boat
So here is the question:
The fuel from a ship leaks into the sea forming a circular oil slick. The area of this circle is increasing at the rate of $20$ $m^2$ per minute.
They asked me to prove that ...
- 1,190
2
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2
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Line $y = mx$ through the origin that divides the area between the parabola $y = x-x^2$ and the x axis into two equal regions.
There is a line $y = mx$ through the origin that divides the area between the parabola $y = x-x^2$ and the x axis into two equal regions. Find m.
My solution:
When I compute my answer, I get $1-\frac{...
- 1,927
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2
answers
295
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Can we clarify this "accumulated money flow" application of integration?
I read about this model/application in Calculus with Applications, 11th Edition by Lial, Greenwell, and Ritchey (example), where if you have a function $f(t)$ that models some revenue stream, the rate ...
- 19k
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35
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What is the fixed "p" percentage I should increase my investments every month to reach a target
I have been learning about SIP. The gist of it is that you invest on regular basis like monthly or quarterly.
The basic example is that you invest 100 every month so it looks like.
...
- 103
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55
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Change in temperature of overflowing container
This is an integation question, form a Physics context.
Mixing of identical fluids at different temperatures is simple, as per here:
https://physics.stackexchange.com/a/24433/290018
We have a slightly ...
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1
answer
85
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What is the variable(s) or such written before the integration symbol?
From 'Distance measures (cosmology)' on Wikipedia:
Cosmologists commonly use the following measures for distances from the observer to an object at redshift $z$ along the line of sight:
Comoving ...
- 151
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320
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Calculus - Calculate Work done to lift water out of tank
I need help setting up the integral so that I can calculate the work done. I've tried it many times and have referred to Youtube, slader, the textbook, and also this site, but I still don't get how to ...
- 133
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2
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85
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Is the area independent of $n$? $ A=\int_0^\infty \frac{e^{-\frac{1}{tn}}(e^{-tn}(tn+1))}{t}~dt $
Consider the parametric equation $x=\frac{1}{t}e^{-tn}$ and $y=te^{-\frac{1}{tn}}.$ To integrate under this curve I put it in the proper form:
$$ A=\int_0^\infty \frac{e^{-\frac{1}{tn}}(e^{-tn}(tn+1))}...
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1
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183
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How to get the integral of $\log(\det(A + Bt))$ w.r.t variable t?
Suppose we have two positive definite matrices $A$ and $B$, now I want to get the integral of:
\begin{align}
\int_{a}^{b} \log(\det(A + Bt)) dt ~~~~~~~~~~~~\text{for } a, b > 0
\end{align}
...
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-3
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1
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set up an integral when the following functions revolve around the $x$, $y$ and $y=\frac{1}{2}$ [closed]
$y=\sin x$, $y=\frac{1}{2}$, $x=0$
i got the same integral which is
$$
\int_0^{ \frac{5 \pi}{6}} \pi \left(\sin^2(x)- \frac{1}{4}\right) \, \mathrm d x.
$$
Anyone help
- 107
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1
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627
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Dirac Delta Function of a complex variable
Is there a context where delta(x+i*c), where c is a real number, makes sense?
It came up while I was doing an Inverse Fourier Transform, and I failed to appreciate its significance.
Does anyone know ...
user564215
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1
answer
109
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Help with volume integration application problem using Disk or Washer Methods, revolving about x-axis, revolving about y-axis.
I need to find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines: y = $\sqrt {x}$ $y=0$, and $x=3$. A) the $x-axis$ B) the $y-...
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2k
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Moment of inertia of a cone frustum with a cylinder cut out (using integral)
How can I find moment of inertia of this frustum when the mass M is uniformly distributed through the grey region using integration?
The hints said to break the region into two pieces, one with ...
1
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1
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686
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what is the volume generated by rotating the given region.
My professor says the volume generated by rotating the region $\mathscr{R}_2$ about the line $OA$ is $5/\pi$ but I don't see how that could be the answer?
- 107
2
votes
2
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225
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Volume of a circle $x^2 +y^2 \leq 1$ which is revolving around a line $x+y=2$.
I want to compute the volume of a circle $x^2 +y^2 \leq 1$ which is revolving around a line $x+y=2$. Usually I solved problems about solids revolving around axis and non axis horizontal and vertical ...
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1
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Solving time derivative of glycogen dynamics: $17.6{dG\over dt} = 2000 - 13G^2$ [closed]
Can I find G, glycogen level at time t=5, if glycogen dynamics are described by the following derivative:
$$17.6{dG\over dt} = 2000 - 13G^2$$
It's been a long time since I've messed with derivatives ...
- 1
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3
answers
84
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examples of cases showing that knowing the area under a curve really matters ( at the elementary level)
It is often said that integral calculus offers a means to solve the area problem. My question, simply aims at understanding what is the interest of this area problem ( at the most basic level).
...
user655689
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1
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417
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Surface Area of Curve Rotated About X-Axis Problem
The question is:
Find the area of the surface obtained by rotating the curve abotu the x-axis.
$x = \frac13(y^2 + 2)^{3/2}$
$1 \le y \le 2$
I do not understand how they went from $(y^2 + 1)^2$ ...
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