-3
votes
1answer
40 views

Calculus use of integral [on hold]

Assume that the price of a product is at a constant value of $\$100$ per unit or the marginal function is $MR=f(x)=100,$ where $x$ equals the number of units sold $a)\ $ What is the total revenue ...
1
vote
3answers
110 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
4
votes
1answer
83 views

any simple method to do integration?

$$\int_{-2}^{x^{2}-2x}e^{t}.e^{t^2} dt = ?$$ What i did is... on rewriting it , $$\int_{-2}^{x^{2}-2x}e^{t+t^2} dt=\frac{e^{t+t^2}}{t^2/2+t^3/3} $$ and then substituting limits is very long process ...
1
vote
0answers
29 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
3
votes
1answer
47 views

Area of a Curved Surface

Find the area of the part o the surface $z=xy$ that lies within the cylinder $x^2+y^2=1$. I'm not sure how to set up the surface integral to compute this.
1
vote
1answer
25 views

Area of a Paraboloid inside a Cylinder

Find the area of the part of the paraboloid $x=y^2+z^2$ that is inside the cylinder $y^2+z^2=9$. I'm not sure how to set up the integral to compute this. Thanks.
2
votes
2answers
22 views

Question on Green's Theorem

Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$. Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where ...
1
vote
0answers
35 views

Evaluating an improper integral with limits $_{-\infty}^\infty$

When evaluating an improper integral with limits $_{-\infty}^\infty$, why do we need to separate the integral into $\int\limits_a^{\infty} \text{ and } \int\limits_{-\infty}^a$? My homework asked ...
1
vote
2answers
69 views

True or False? $\int\limits_0^2(x-x^3)dx$ represents the area under the curve $y=x-x^3$ from 0 to 2.

True or False? $\int\limits_0^2(x-x^3)dx$ represents the area under the curve $y=x-x^3$ from 0 to 2. I said true but my textbook says false. Why? (Stewart: Concepts and Contexts p424 q13)
3
votes
2answers
55 views

Evaluate integral by completing the square and doing trigonometric substitution

$\int \frac{1}{(x-2)\sqrt{x^{2}-4x+3}} dx$ is my problem Complete the square $\int \frac{1}{(x-2)\sqrt{(x-2)^{2}-1}} dx$ I know I'm probably supposed to use $ \frac{d}{dx}\operatorname{arcsec}(u) = ...
11
votes
3answers
225 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
12
votes
1answer
148 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
3
votes
5answers
197 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
6
votes
3answers
250 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
5
votes
3answers
111 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
6
votes
5answers
185 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
0
votes
0answers
15 views

Partial derivative of straigh-line parametrized integral

I would like to evaluate the following $$ F(\mathbf{r}_1,\mathbf{r}_2) = \int_0^1 ds~f(\mathbf{r}_1 + (\mathbf{r}_2 - \mathbf{r}_1)s) $$ where $\mathbf{r}_{1/2} = (x_{1/2} , y_{1/2})$, i. e. a ...
1
vote
3answers
37 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
1
vote
1answer
41 views

Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...
0
votes
1answer
14 views

Integral, left-hand sum

Could anyone explain why my first answer is wrong? what I did was delta x = 10/5 = 2 $$ 2(2^2+1)+2(4^2+1)+2(6^2+1)+2(8^2+1) = 248 $$ and the second answer was $$ ...
1
vote
1answer
43 views

How to properly generalize a definite integral?

I know, I know. On the can, this problem seems simple. Just take $\int_a^bf(x)\mathrm{d}x$ and write is as $\int f(x)\mathrm{d}x$. However, when I tried to do that on an Engineering Dynamics ...
1
vote
1answer
38 views

Arc Length in two dimensions by integration

I'm really at the end of my wits on this problem. Basically I'm trying to find arc length. The vector-valued function is: $R=\langle t,\sqrt{t}\rangle$ and $t\ge0$. We're looking for the length of ...
1
vote
3answers
386 views

The high power integral

Im trying to solve the indefinite integral $$\int\frac{x}{(x^2+4)^3} \, \mathrm{d}x $$ I tried applying polynimial division and breaking to partial fractions but it didnt help...are there any other ...
2
votes
2answers
36 views

Proove of equality of integrals

I'm currently sitting on the following problem: Let f be in the set of the integrable functions(:=$L^¹(\mathbb{R}^n))$, A $\in \mathbb{R}^{n\times n}$ invertible. Therefore define g:=$\mathbb{R}^n ...
4
votes
1answer
86 views

Solving for limit of integration

$$\frac{1}{\sqrt{2\pi}} \int^0_{z_a} e^{\frac{-z^2}{2}} \, dz = 0.48 $$ How would I solve for the value of $z_a$ using a calculator?
9
votes
2answers
109 views

Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
0
votes
0answers
33 views

Estimation of solution to $u_t=u_{xx}+x^3u_x$ using integrals

Let $$u_t=u_{xx}+x^3u_x$$ With: $$u(0,x)=u_0(x)$$ $$u(t,0)=u(t,l)=0$$ Find an energy approximation of $u$ on $(0,T) \times(0,l)$. By multiplting by $u$ we get: ...
2
votes
2answers
52 views

Finding the length of a spiral

I need to find the length of a spiral. The spiral start at a certain radius $R_1$ and ends at a smaller radius $R_2$. As the spiral spins inwards, the distance between each arm of the spiral decreases ...
0
votes
1answer
28 views

Using Midpoint Rule to Approximate a Definite Integral

I got this question wrong.. I started by obtaining the following sample points $-.7, -.3, .1, .5, .9$ Next I got my $\Delta x$ with the following computation $\displaystyle{\frac{1.1 - (-.9)}{5} = ...
0
votes
1answer
26 views

Using the Weibull Distribution, derive $E(X^k)$

If $X$~WEI$(\theta,\beta)$, derive $E(X^k)$ assuming $k\gt-\beta$. Note that $X$~WEI$(\theta,\beta)=\frac{\beta}{\theta^{\beta}}x^{\beta -1}e^{-({x}/{\theta})^{\beta}}$ I am having a very difficult ...
0
votes
2answers
70 views

I Don't Understand Error Bounds

I understand they're supposed to give us a limit on how off our approximation of an integral can be, but I don't understand how the formula gives that. What does the second derivative have to do ...
3
votes
1answer
116 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
2
votes
1answer
58 views

How to calculate this integral using Rodrigues' formula?

I'm trying to get practice using Rodrigues' formula for Legendre Polynomials, but it's being quite confusing to manipulate that $n$-th derivative. Basically, I'm trying to calculate: $$\int_{-1}^1 ...
2
votes
3answers
106 views

How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
1
vote
2answers
38 views

Integration related question

how does one integrate $ (x^3-1) /( 4x^3-x) dx$ ? I tried dividing polynomials but it didnt help....
1
vote
2answers
35 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
-2
votes
2answers
42 views

Fourier transform of t*(sent/pi*t)^2 [on hold]

Here's the function (I need it's fourier transform).
0
votes
2answers
61 views

Evaluation of a definite integral involving logarithms and square roots

I really don't understand this, please someone help me!
-1
votes
0answers
15 views

Line integral with vector function

I have the following question, to which I got close, but eventually sought help online. Evaluate the line integral $\int_C F \dot{} dr$ where $F(x,y,z) =\langle2\sin(x) , -3\cos(y),4xz\rangle$ and ...
2
votes
4answers
88 views

integration by parts of trig functions

Can anyone help me with this integral? $\int{x^3 \sin(x^4) dx}$ I set $u=x^3$, and I let $v=-\cos(x^4)$, so that $\frac{dv}{dx}=\sin(x^4)$ I tried using integration by parts, but, whenever I come ...
3
votes
5answers
113 views

Evaluate $\int \sqrt{1-x^2}\,dx$

I have a question to calculate the indefinite integral: $$\int \sqrt{1-x^2} dx $$ using trigonometric substitution. Using the substitution $ u=\sin x $ and $du =\cos x\,dx $, the integral becomes: ...
0
votes
1answer
41 views

Volume bounded by two cylinders

I'm confused by these "surfaces". I know how to find volume when curve rotates around $Ox$ or $Oy$ line, but I don't know what to do with surfaces. I need to find volume bounded by surfaces $$ \left(x ...
-1
votes
0answers
27 views

problem on partial integration in two variables

I am trying to find the solution to an assignment, the question is, to find the solution to the following fourth order differential equation in two variables; $D^4w/Dx^4+2D^4w/Dx^2Dy^2+D^4w/Dy^4=q/k$ ...
3
votes
4answers
150 views

How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
2
votes
2answers
30 views

About the order of integration for double integrals

I have to compute $\int\int (2x-y) \,dx \, dy $ on the domain $\{ (x,y) \in R^2 : 1\leq x\leq 4, 0\leq y\leq \sqrt{x} \}$ So mi first try is to do: $\int_0^{\sqrt{x}}\int_{1}^{4} (2x-y)\, dx \, dy ...
0
votes
1answer
15 views

Center of Mass double Integral using polar Coord.

Find center of mass given Lamina pictured: https://s3.amazonaws.com/wamapdata/qimages/qtrring.gif with inner radius of 3 and an outer radius of 7, and a density function $$\rho(x,y) = ...
0
votes
1answer
67 views

physics math problem with dirac delta

I'm trying to do a homework problem where they have asked me to recover the usual coordinate space momentum operator from its hilbert space equivalent. it gets you to show: ...
2
votes
2answers
85 views

convergence of a generalized Riemann integral

Could you please provide me some hints to test the convergence of this integral below? $$\int\limits_{0}^{+\infty}\dfrac{\sin x\cdot \sin 2x}{x^\alpha} \, dx$$ where $\alpha \in \mathbb{R}$
1
vote
3answers
39 views

Proving Definite Integral Strong Inequation

I got the following question: Prove: $$\int^{1}_0 \frac{x^9}{\sqrt{1+x}} < \frac{1}{10}$$ I proved the weak inequation like so: $x \geq 0 \Rightarrow \sqrt{1+x} \geq 1$ (monotonicity of square ...
1
vote
2answers
67 views

Definite Integration: $\;36\pi \int_{0}^{5}t^3\sqrt{t^2+1}dt$

$\displaystyle 36\pi \int_{0}^{5}t^3\sqrt{t^2+1}dt$ I'm in Single Variable Calculus. This problem is about finding a surface area of parametric equations, and I have substituted values in. But I ...