1
vote
0answers
35 views

Finding the mean value of y

I don't understand how to obtain the limits for the $t$-values considering that they gave us the $x$-values in the first part of the equation. I've considered substituting the $x$-values into the ...
6
votes
2answers
140 views

Find the length of the curve $x^{2k}+y^{2k} =1$

I want to find an expression for length and find the limit $k\rightarrow \infty$ The answer is obviously 8, if we look at the graphs.
1
vote
1answer
28 views

Area of the surface generated by revolving curve around y-axis

So I did something wrong in my solution because I'm not seeming to get the right answer. $$\int_c^d 2\pi (4 \sqrt{9-y}\sqrt{1-\frac{4}{9-y}})~\mathrm{d}y$$ combine square roots and move out ...
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
1answer
29 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
0
votes
1answer
39 views

Prove the continuity and differentiability of parametric integration

$$F(\alpha )=\int_{0}^{+ \infty } \frac{\cos x}{1+(x+\alpha )^{2} } dx$$ Prove the function F is continuous and differentiable on the interval $[0, +\infty )$
3
votes
1answer
52 views

Area of part of parametric function

I need to get area of function: $x= 2\sqrt{2}\cos ^3 t$ and $y= 4\sqrt{2}{\sin ^3 t}$, but only the part when $x\geq1$. How can I do that? I know that area of full function would be $$S= \int_a^b ...
4
votes
1answer
42 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
0
votes
2answers
78 views

Paramtrizing a counterclockwise circle vs. a clockwise one

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? For example, I am looking at computing an integral $\int_\gamma ...
0
votes
0answers
42 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
1
vote
0answers
39 views

Line integral, Parametrization

I have this line $A=\{(x,y) \in R^2 : y^2+4x^4-4x^2=0\}$ , $(x>0)$ I parametrized it like that : $b(t) = (t, \sqrt{4t^2- 4t^4})$. And my $F$ is $F(x,y) = (x+y,-x)$. But when I calculate my ...
0
votes
1answer
146 views

Surface Area of a Parametric Curve

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. $$ ...
1
vote
1answer
75 views

area under a parametric curve problem

so I have a parametric curve, x = cos(t) y = sin(2t) I found that I need the area from 0 to pi/2. put this into an integral in terms of t I get $$ -\int_0^{\pi/2}sin(2t)sin(t)dt $$ But in my ...
0
votes
1answer
76 views

Some questions about parametric integrals

1) What is the error in the following calculation ? $\int_{0}^{oo} \frac {sin(px)}{x}dx$=$\frac {\pi}{2}$ derivating by p at both sides $\int_{0}^{oo} cos(px)dx$=0 But the second integral does not ...
4
votes
1answer
291 views

Using geometric arguments to solve an analysis problem

Im not good in geometric interpretations... any help is very welcome. Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$ parameterized by ...
1
vote
1answer
66 views

Parametric representation of $\sqrt{x^2+y^2}\le z \le 2$

just wondering how to parametrize this. Question is: Let $C$ denote the conical region $\sqrt{x^2+y^2}\le z \le 2$. Find a parametric representation $\mathbf{x}(u,v)$ for $S$, the surface of $C$. ...
0
votes
2answers
399 views

Explanation of the area under the curve given by a parametric equation

My textbook says the area under a graph is given by: $\smallint ydx$ And it then goes on to say by the chain rule: $$\smallint ydx = \smallint y{{dx} \over {dt}}dt$$ Could someone explain to me how ...
4
votes
2answers
491 views

Finding surface area of a cone

I will describe the problem then show what I tried to solve it. I need to find the area of the cone defined as follows: $$z^2=a^2(x^2+y^2)$$ $$0\leq z\leq bx+c$$ where $a,b,c>0$ and $b<a$. ...
1
vote
1answer
128 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
3
votes
2answers
102 views

Having trouble solving question involving parametric equations

I have been given the following: $$y = a \cdot \cos^3t$$ $$x = a \cdot \sin^3t$$ $$0 \leqslant t \leqslant {\frac\pi2}$$ I am supposed to show that the mean value of $y$ over the interval ...
0
votes
1answer
57 views

Line integrals of vector fields

Consider the vector field:$$\vec G = \left(\frac y{x^2+y^2}, \frac {-x}{x^2+y^2}\right)$$ compute $\int_\Gamma \vec G$ where $\Gamma$ is the proportion of a parabola $y=a(x-1)^2$ from (1,0) to (2,a). ...
1
vote
0answers
106 views

Computing the surface area of a (piecewise) polynomial parametric surface

I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases. Case ...
2
votes
2answers
303 views

parametric curves, parameter and integration

I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between ...
0
votes
1answer
409 views

Help me understand a surface integral question?

The question is: Evaluate the surface integral: $$ \iint\limits_S \, x^2yz\ \mathrm{d} S $$ Where S is part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3] X [0,2] I literally just ...
2
votes
1answer
231 views

Finding parametric equations

I am trying to understand volume and surface integrals. I do get the idea of the process (find a parametric equation of the volume/surface, integrate afterwards). But I just cannot make up parametric ...
1
vote
1answer
118 views

Find the Frenet frame

Consider the following space curve: $$ \gamma(x)=(e^x\cos(x), e^x\sin(x), e^x). $$ My main goal is to find the Frenet Frame T,N,B. So far I have found the arc-length using the following formula: $$ ...
1
vote
1answer
144 views

Range of Parameters and Integral Evaluation

I am currently working on understanding this problem and am in need of some assistance. I'll let you know what I've done so far, and then hopefully someone will be able to help me. So the problem is ...
1
vote
2answers
350 views

Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation?

I have the following parametric equation: $$x=t^2-2t$$ $$y=\sqrt{t}$$ I'm interested finding the area of the region bounded by this curve and the y-axis (i.e. $0 \leq t \leq 2$). We have: ...
3
votes
1answer
321 views

Solving integral equation with Laplace's Transform.

I'm trying to prove the following $$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$ The original problem ...
1
vote
1answer
3k views

Find a parametrization of the curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$ and use it to compute the area of the interior.

I have the following homework question that I know the answer to $(3\pi/8)$, however, I don't understand how to get this answer. The question: Find a parametrization of the curve $x^{\frac{2}{3}} + ...
1
vote
1answer
496 views

Parameterization of an implicit function

I'm trying to find the area of an irregular domain that is bounded by $x = c$, $y = c$, and $c = -A\sin(x/2)\sin(y/2)+\cos(x/2)\cos(y/2)$, where A can vary in the range [-1,1], and x and y are only ...