Mathematics Stack Exchange
Mathematics Stack Exchange

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
0
votes
1 answer

Given $f$ bounded. $\forall x \in [a,b]: f(x)\geq 0 $ and $\exists c \in [a,b]: f(c)>0, $ with $f$ continuous at $c$. Show that $\int_{a}^{b} f(x) >0$

Are these conditions sufficient to demonstrate integrability of the function? Or did I miss writing down integrability as hypothesis
Marco
  • 15
0
votes
1 answer

Integrating $\int_0^{2} \left(1+x^3\right)^{1/2}+\left(x^2+2x\right)^{1/3} dx$

I stumbled upon this question while solving an exercise of DI. $$\int_0^{2} \left(1+x^3\right)^{1/2}+\left(x^2+2x\right)^{1/3} dx$$ I observed something... If we denote $f(x)=(1+x^3)^{1/2}$, then the second part of the integrand is merely…
Nex
  • 27
0
votes
1 answer

Convergence of $\int_b^\infty \left(\sqrt{\sqrt{x+a} - \sqrt{x}} - \sqrt{\sqrt{x} - \sqrt{x-b}}\right)dx$

For what pairs $(a, b)$ of positive real numbers does the improper integral \begin{equation} \int_b^\infty \left(\sqrt{\sqrt{x+a} - \sqrt{x}} - \sqrt{\sqrt{x} - \sqrt{x-b}}\right)dx \end{equation} converge. I conjecture that $a = b$, but I'm not…
Math_Day
  • 1,227
0
votes
1 answer

What is wrong with this "derivation" of the Gaussian integral?

Let $g(z) = \sqrt{z}$, so that $g^{-1}(z) = z^2$ and $Dg^{-1}(z) = 2z$. Using the change of variables formula, we have that $$\int_0^T e^{-z^2/2}\ dz = \int_0^{T^2} e^{-z/2} 2z\ dz.$$ We can compute $\int 2ze^{-z/2}$ by differentiating under the…
TheProofIsTrivium
  • 1,337
0
votes
1 answer

Evaluating $ \int_0^\infty \int_0^\infty(v_1 v_2)^2e^{-a(v_1^2+v_2^2)}\;|v_1 - v_2|\;d v_1 d v_2$

I was doing a physics calculation and got stumped on this. What is the solution of this integral? $$ \int_0^\infty \int_0^\infty(v_1 v_2)^2e^{-a(v_1^2+v_2^2)}\;|v_1 - v_2|\;d v_1 d v_2$$ where $a$ is a positive constant. I'm pretty confident it…
More Anonymous
  • 1,183
0
votes
1 answer

Evaluating $\int_{0}^{\pi / 2} \ln \left(1+3 \sin ^{2} x\right) d x$

I need a hint for evaluating $$\int_{0}^{\pi / 2} \ln \left(1+3 \sin ^{2} x\right) d x$$ I've stumbled upon this integral which is from a calculus textbook of mine and I have no idea how to solve it. I don't see how a substitution would do the work…
user571454
0
votes
1 answer

Equation of a variable plane passing through the point $(a,b,c)$

If we are to write the equation of a variable plane that passes through the point $(a,b,c)$ then can we say the general equation of the plane will be $\dfrac xa+\dfrac yb+\dfrac zc=3$? Or would it be some specific plane only and not a general one? I…
aarbee
  • 8,246
0
votes
0 answers

Why do we use a right triangle as the approximation when calculating area in polar coordinates system?

This question comes up when I read the Page 4 of MIT Mathematics OpenCourseWare 18.01 lecture note. In a polar coordinates system, when calculating the area of the function $r = f(\theta)$, why did the lecture note choose a "right triangle" to…
Dachuan Huang
  • 151
0
votes
0 answers

Double Integral (simple question)

This is a simple exercise on multiple integration. I am given with $$R=\left \{ (x,y) \in \mathrm{R}^2 \mid y^3 \leq x \leq y, 0 \leq y \leq \sqrt{\frac{\pi}{3}} \right \}$$ and asked to evaluate $$ \iint_R 2\sin y^2 \; dA. $$ I just wanna know if…
user50605
  • 31
0
votes
1 answer

Compute the following integral $\iiint _D xydzdydx$

Compute the following integral $$\iiint_D xydzdydx$$ Where $D$ is the space region restricted to $z=4-x^2-y^2$ and $x^2+y^2=1$ and $z=0$. Here is a plot: So I think the triple integral is…
masaheb
  • 918
  • 6
  • 16
0
votes
1 answer

Definitive Integral, regarding cross sections and colume

The base of a solid is bounded by y=x^1/2, x=1, x=4, and y=0. What is the volume if the solid has square cross sections perpendicular to the x axis? I got 15pi/2, but I'm not sure that's right. Could anyone help me find the right answer? the cross…
Julia
  • 21
0
votes
1 answer

Calculate definite integral-2

Why integral $I(b,x) = \int\limits_x^\infty {{e^{ - ({y^2} + by)}}dy = \sqrt \pi {e^{{b^2}/4}}N( - \sqrt 2 - b/\sqrt 2 )} $ where the function $N$ is defined as $N(x)=\frac{1}{{\sqrt {2\pi } }}\int\limits_{ - x}^\infty {{e^{ - {y^2}/2}}dy = }…
Adam
  • 3
0
votes
1 answer

limits of an integral with delta

Having this integral: $$\int_{x-t}^{x+t}\delta(z)dz$$ the solution is: $$ \begin{cases} 1& \text{for $|x| < t$} \\ 0 & \text{else} \end{cases}$$ Namely, all the domain of…
Danny Dan
  • 61
0
votes
1 answer

Definite integral between variables

I am trying to understand the book, when it says that: "Integrating between time 0 and time T, we get.." What I need to integrate is this: $\frac{d S}{S}=\mu d t$ And when this is integrated between 0 and T, the book states that we get: $S_{T}=S_{0}…
mbih
  • 37
  • 7
0
votes
0 answers

Calculating velocity and position with integrals

A ball is dropped from a height of 3000 feet. Its velocity after t seconds is $v(t)=-78t$. a) How fast is the ball dropping after 5 seconds? b) Determine the position function $s=s(t)$. For a), I multiplied $64$ by $5$ and got $320$. But, I'm…
Julia
  • 21
1 2 3
…
76 77