Homework Statement
Evaluate the triple integral \int\int\int^{}_{E} xy dV where E is the tetrahedron (0,0,0),(3,0,0),(0,5,0),(0,0,6).
Is there a simple way to simplify the integration?
Homework Equations
The Attempt at a Solution
\frac{z}{6} + \frac{y}{5} + \frac{x}{3} = 1
z =...
Didn't even think about that . . .but for some reason it doesn't seem to be working.
I switched the integration around and got the following.
\int_0^1 \int_0^4 5ye^{-xy} dxdy
After the first integration it comes out really neatly as
-5e-xy
and after plugging in 0 and 4 I get
-5e-4y...
Homework Statement
\int\int\int^{}_{B} ye^(-xy) dV where B is the box determined by 0 \leq x \leq 4, 0 \leq y \leq 1, 0 \leq z \leq 5.
Homework Equations
The Attempt at a Solution
\int^{4}_{0}\int^{1}_{0}\int^{5}_{0} ye^(-xy) dzdydx
Integrating the first time I get
zye-xy
Plugging in 5...
Okay, then after integration I get the following.
128*pi/2 - [1/2\theta + sin(2*\theta)/4]
Plugging in my values I get the following.
128*pi/2 - pi/4 - sin(pi)/4 + sin(0)/4
but this is wrong. I am not sure what I am doing wrong.
That seems to make sense since if I think of \theta as how much an imaginary line sweeps through and radius as how long that line actually is.
So now that I have the following:
\int^{\pi/2}_{0}\int^{16}_{0} r drd\theta - \int^{\pi/2}_{0}\int^{16cos\theta}_{0} r drd\theta
After the first...
I sketched it out and for the second one I am basing my intervals of integration of r from the center of the circle to the edge of the circle.That seems to make the most sense to me, however, a similar example in my book (which has a circle shifted 2 to the right instead of 8 and has a radius of...
Homework Statement
Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x2 + y2 = 256 and x2 - 16x + y2 = 0.
Homework Equations
The Attempt at a Solution
Finding the intervals of integration for the polar coordinates...
Oh I am so sorry about that, I completely mistyped that. It was supposed to be dx/du, dx/dv, dy/du, dy/dv. Also thanks you pointed out where I got it wrong. That helped a lot.
I think I calculated my Jacobian correctly since I am taking the correct partial derivatives of my functions.
dx/du -1/7u + 3/7v = -1/7
dx/dv -1/7u + 3/7v = 3/7
dy/du 2/7u - 3/7v = 2/7
dy/dv 2/7u - 3/7v = 3/7
-1/7*2*7-3/7*3/7 = -2/49-9/49 = -11/49 and absolute value gives me 11/49
Homework Statement
Suppose D is the parallelogram enclosed by the lines 2x-3y = 0, 2x-3y = 2, 3x-y = 0 and 3x-y = 1.
\int\int^{}_{D} [(2x-3y) e^(3x-y) dA
Homework Equations
The Attempt at a Solution
Set u to be equal to 2x-3y -> x = (u+3y)/2
Set v to be equal to 3x-y -> v = 3/2(u+3y)-y...
Homework Statement
Suppose D is the parallelogram in the xy-plane with vertices P(-1,5), Q(1,-5), R(5,-1), S(3,9)
\int\int ^{}_{D} (6x+12y) dA
HINT: Use transformation x = \frac{1}{6}(u+v) and y = \frac{1}{6} (-5u+v).
Homework Equations
The Attempt at a Solution
Calculating the Jacobian I...
Homework Statement
Evaluate the integral by reversing the order of integration.
\int^{3}_{0}\int^{9}_{y^2} y cos(x^2) dydx
Homework Equations
...?
The Attempt at a Solution
Drawing the picture out we get a sideways parabola.
From the picture I get the following intervals of integration...
Homework Statement
Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 9.
Homework Equations
. . . ?
The Attempt at a Solution
After drawing out the picture with z=0 I have a line going from 0,9 to 9,0 bounded by the x and y axis giving me a triangle...
Homework Statement
Evaluate the iterated integral I = \int^{1}_{0}\int^{1+y}_{1-y} (6y^2+ 10x) dxdy
Homework Equations
. . . ?
The Attempt at a Solution
Integrate with respect to x gives me the following equation.
\int^{1}_{0} 6xy^2 + 5x^2 dy
I plug in y+1 and y-1 into x and get the...
Whoops, I forgot to add in a step. However, when I integrate that I get what I have shown in my question and when I evaluate it on the interval from 0 to 6 I am getting the wrong answer which makes me think that maybe I am doing something else wrong.
Homework Statement
Evaluate the double integral ∫∫D xy dA where D is the triangular region with vertices (0,0) (6,0) (0,1).
Homework Equations
The Attempt at a Solution
0 <= x <= -\frac{1}{6}x+1
0 <= x <= 6
the first integral would be the integral from 0 to -1/6x+1 of xy with...
I don't think that would be it since the cross product formula says that for the middle number it is subtraction. Besides, it is being squared, even if it is negative the answer which results will be positive.
Homework Statement
An implicit equation for the plane passing through the point (-2,5,-5) that is perpendicular to the line L(t) = <5+2t,3,4> is ...?
Homework Equations
a(x-x0) + b(y-y0) + c(z-z0) = 0
The Attempt at a Solution
So in order to find the equation of the plane I would...
Homework Statement
Find an equation of a plane containing the three points (1, -1, 0), (5, 4, 1), (5, 5, 3) in which the coefficient of x is 9. What does it mean when it says the coefficient of x is 9?
Homework Equations
a(x-x0) + b(y-y0) + c(z-z0) = 0
The Attempt at a Solution...
In response to LCKurtz, sorry but I don't see how it works conceptually. What it looks like to me is that you are getting the magnitude of the cross product on the direction vector of the line and the line from the line to Q. From what I understand about cross products is that this will give us...
Homework Statement
Find the distance from the point (2, 4, 4) to the line x = 0, y = 4 + 3t, z = 4 + 2t.
Homework Equations
The cross product and the dot product and d = |n * b|/|n|
The Attempt at a Solution
So the distance from the point to the line is the line directly...
Homework Statement
A spherical source radiates sound uniformly in all directions. At a distance of 9 m, the sound intensity level is 100 dB. What power is radiated by this source?
Just a simple answer check to see if my answer is reasonable.
Homework Equations
\beta = 10 db log (I/I0)...
Homework Statement
A raft is made of 11 logs lashed together. Each is 35 cm in diameter and has a length of 5.6 m. How many people can the raft hold before they start getting their feet wet, assuming the average person has a mass of 70 kg? Do not neglect the weight of the logs. Assume the...
Homework Statement
So the original problem goes as the following.
A uniform ladder of mass (m) and length (L) leans against a frictionless wall, see figure. If the coefficient of static friction between the ladder and the ground is 0.41, what is the minimum angle (q) between the ladder and...