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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.

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What part of area is included in the definite integral?

I am supposed to find the area of the blue shaded region. The line $\mathrm {OA}$ is $y=x$. The circle is $x^2+y^2=16$. The best method is to find slope of the line and use the formula $\frac12r^2\theta$ for the area of sector. But I'm supposed to…
Tejas
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What domains and codomains has the definite integral been defined for?

I'm often interested in generalizing functions, and one of the things I was thinking about recently was the concept of the Riemann definite integral as a function with some rather specific domains. It made me think and wonder about the…
dezakin
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Hint for the integral

I was trying to compute the following integral but got stuck at the starting point. Can anyone provide a valuable hint for the evaluation of this integral $\int_{0}^{\infty}x^{9}e^{-x^{2}} dx$ ?
bpr3003
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Area of $f(x,y) = \frac{1}{(2x+3y)^2}$

I was asked to find the area of $f(x,y) = \frac{1}{(2x+3y)^2}$ inside the parallelogram defined by the points (5,-1); (8, -3); (6, -1) and (9, -3). I am stuck trying to find a suitable change of variables in order to integrate inside of a rectangle.…
John
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shell method not giving right answer

The soot produced by a garbage incinerator spreads out in a circular pattern. The depth, $H (r)$, in millimeters, of the soot deposited each month at a distance $r$ kilometers from the incinerator is given by $H (r) = 0.116 e^{-1.5 r}$. I'm supposed…
Ahmed Al.Mutaiwea
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Table of Integrals. Which one to use?

I just finished my exam and there was this question. It asks us for which form in the table of integrals to use. $\int(x-3)\sqrt(6+6x-x^2)$ I did completing the square and got into this form $\int(x-3)\sqrt(15-(x-3)^2)$ then u-sub with u= x-3 and I…
Zhi J Teoh
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find the derivative of the function g(x)

Find the derivative of $$g(x)= \int_1^{\cos x} \sqrt[3]{1-t^2} \ dt$$ I am having trouble finding the integral. I thought you would set u as 1-t^2 but it doesn't work
blake
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can't solve this definite integral: $4\pi \int_1^e \frac{\ln x}{ x} \; d x$

please help me with this integral, I can't figure out how to solve it from the manual. $$ 4\pi\int_1^e \frac{\ln(x)}{x} \, \mathrm{d}x $$
Mihai101
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Definite integral by substitution

Why isn't the answer (d) but (b)? Thank you for your help!!
Emily
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Help find mistake in $\int_0^{π/4} (\cos 2x)^{3/2} \cos(x) \, \mathrm{d}x$

$$ \int_0^{π/4} (\cos 2x)^{3/2} \cos(x) \, \mathrm{d}x $$ I was able to find out the correct answer as $\frac{3\pi}{16\sqrt{2}}$ using $t = \frac{\sin\theta}{\sqrt{2}}$ instead of above. But I want to know what mistake I might have done in above…
Daksh
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find the point of maximum function

can't figure out how to solve this, completely forgot the math course Could you point by point how to do this? \begin{gather}\int_0^x e^{-t^2} . (t^2-17t+72) \,dt\end{gather} enter image description here
Денис Денис
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what is $\int_{1}^\infty \frac{x!}{x^x}dx$?

I plugged in this integral on desmos $$\int_{1}^\infty \frac{x!}{x^x}dx$$ and it said it was undefined. I see no reason this should be undefined because $\frac{x!}{x^x}$ goes to 0. What is this integral or is it just undefined?
user1025147
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how to write $\iiint_R (x^2+y^2+z^2) dV $ where $R=\{x^2+y^2+z^2\le4, 1\le z\le2\}$

How to write $\iiint_R (x^2+y^2+z^2) dV $ where $R=\{x^2+y^2+z^2\leq 4, 1\leq z\leq 2\}$ the limits? I know that $1 \leq z\leq 2$ but for $x, y$ what should be? Attempt: $y=\sqrt{4-z^2-x^2}$ or $x=\sqrt{4-z^2-y^2}$.
apa
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Integral $\int^{x=\frac{\pi}{4}}_{x=0} \int^{y= \cos{x}}_{y=\sin{x}} dydx$

$$\int^{x=\frac{\pi}{4}}_{x=0} \int^{y= \cos{x}}_{y=\sin{x}} dydx$$ I got the answer $\sqrt{2} - 1 $ but my tutor got $8$? I assumed that I am starting with integrating $1 dydx$, that is how I got my answer. Is my answer wrong?
Lils
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Integral of double sin

Consider $n\in\mathbb{N}$ and $l\in\mathbb{N}$. How do I calculate the integral that follows? $$\int_0^L x\sin\left(\frac{\pi n x}{L}\right)\sin\left(\frac{\pi l x}{L}\right) dx$$
Luis Mora
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