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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

20559 questions
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how to solve this definite integrals?

$$\int_{a}^{b} \frac{\sqrt{(r-a)(b-r)}}{r}dr$$ where a and b are constant lower and upper limit. The answer of this integral is $$\pi/2({a}^{1/2}-{b}^ {1/2})^{2}$$ so please give me the hints that how to solve this integral thanks in advanced sir.
umairkhan
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double integral of bi-variate normal pdf

Just want to confirm this. It's past mid-night and my brain is not working. LEt $\Phi(x,y)$ and $\phi(x,y)$ be the CDF and PDF of bi-variate normally distributed variables. Then $\int_{b_{i-1}}^{b_i} \int_{a_{i-1}}^{a_i} \phi(x,y) dx dy =…
Amatya
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Definite Integral of $\sin 2\pi x $

I'm trying to find: $$\int_0^{1} \sin 2\pi x \; dx =$$ My friend says the result should be zero. Is there any graphical way to prove it.
Gin99
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Double iterated integration: what am I doing wrong?

The question is: integrate $f(x,y) = 1$ on the region bounded by: $$0 \le y \le -x^2 + 1$$ $$-1 \le x \le 0.5$$ I've turned this into the following integral: $$\int_0^1 \int^{0.5}_{\sqrt{1-y}} 1 dx dy$$ When I evaluate this, I get $\frac{1}{6}$. …
Michael Stachowsky
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Integral technique question

For the first part of the question I can calculate the final answer as $$\frac{2\tan^{-1}\sqrt{\frac{y-1}{y+1}}}{\sqrt{y^2-1}}$$ However I am still unable to see how to get the second part. I tried differentiating the first one with respect to $y$…
DJA
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Homework Problem on Integration

$g(x)= \tan(x)+ 2\tan 2x + 4\tan 4x + 8\cot 8x$ and $f(x) = (7\tan^{6} x -3\tan^{2} x) \dfrac{1}{g^{2}(x)+1}$; $x \in \left( 0,\dfrac{\pi}{4} \right)$ Then find $\displaystyle \int_0^\frac{\pi}{4} f(x) dx$ and $\displaystyle \int_0^\frac{\pi}{4}…
Abhash Jha
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A simple definite integral

It has been a while since calculus and I am having a hard time remembering how to solve this problem. $\int_{0}^{x} 9e^{-3x}dy$ I know that the answer is $9xe^{-3x}$, but I am not sure what the steps are.
Alex
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A very simple question on definite integrals

The original question was " Integrate $\sqrt{1+\cos x}$ from $0$ to $200\pi$ " , so what I am in doubt is that in area we don't cancel positive and negative areas, but in definite integrals it should come out to be $0$ but in the solution, the…
satyatech
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Integral of the multiplication of two funtions

I know that $f(x)$ and $g(x)$ are two continuous functions over $x>0$. If $\int_0^\infty f(x) dx>0$ and $g(x)>0$, $\forall x>0$, under what conditions $\int_0^\infty f(x)g(x) dx>0$? Can we say anything in general?
Salivan
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Error approximation bound of using trapezoidal rule?

Suppose you wish to approximate $\displaystyle \int_{1}^{4} \ln(x)dx$ by using Trapezoidal rule. What value of n would you chose to ensure that the error in your approximation is no more than $10^{-2}$. Using: $$|E_{T}| ≤ \frac{K_{2}(b -…
yre
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How to Find Definite Integral Superfast

Compute $\int_{-2}^6 (8x^5-6x^3+7x-3-22\sin(\pi x))\,dx$. Simple question: When I take the indefinite integral of this in $0.1$ seconds, is there a way to avoid spending the next $2$ hours crunching through arithmetic?
Saketh Malyala
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I want to know the proof of the definite integral :

$$ \int _{0}^{\infty }\frac {x^{p-1}dx}{1+x}=\frac {\pi }{\sin(p\pi )}$$ for $0
Sayandip Banerjee
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Integral of x ln(x)

$$\int_0^1{x\ln(x)dx}$$ I tried to evaluate it but I got stuck at this part - $\lim\limits_{x\mapsto 0}\log(x)$ I believe that is limit doesn't exist and hence it is impossible to evaluate the integral. Am I correct?
Dev Aggarwal
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How the average value is computed for this sine wave

In the following picture, I'm trying to compute the average value of the blue wave which represents the output of a half-wave rectifier circuit. In the book, it is $V_o \approx (1/\pi)V_s - V_D/2$ where the black wave is a sine wave. I'm not sure…
CroCo
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Sketch region and get definite integration

I'm teaching myself definite integration and found a question like this. I have to sketch the region and then find the area through definite integration. Can anyone help? The following is the question. Sketch the region enclosed by the line $2x +…
Jessie
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