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course Mth 277
4/15 1
Question: `q001. Evaluate the double integral Int( Int(dr, 0, 1 + sin(theta)) dθ), 0 , pi). Sketch and describe the region of integration.In standard notation the integral is
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Your solution:
Int( 1 dr) = r, evaluated at limits
[1+sin(`theta)] - 0 = 1+sin(`theta)
Int(1+sin(`theta) d`theta) = Int(1 d`theta) + Int(sin(`theta) d`theta)
=`theta - cos(`theta), evaluated at limits
[`pi - cos(`pi)] - [0 - cos(0)]
=`pi + 2
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Given Solution:
An antiderivative of 1 is r. The change in the antiderivative between r = 0 and r = 1 + sin(theta) is 1 + sin(theta).
Integrating this result with respect to theta we get antiderivative theta - cos(theta).
The change in this quantity between theta = 0 and theta = pi is
pi - cos(pi) - (0 - cos(0) ) = pi - (-1) - (-1) = pi + 2.
Additional note:
Between theta = 0 and theta = pi the value of sin(theta) goes from 0 to 1 and back to 0, so the value of 1 + sin(theta) goes from 1 to 2 to 1.
The graph of 1 + sin(theta) from theta value 0 to 2 pi is given below, and the portion of the graph from theta values 0 to pi is in light blue, the portion above the x axis. It should be clear that this region contains a circle of radius 1 (visualize the circle centered at (0, 1)) and therefore has somewhat more area. The area of the circle would be pi, so the result pi + 2 makes good sense.
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Self-critique (if necessary):
Still can’t see the graphs the best but getting better. So we only have the form Int(Int( “”” r dr) d`theta) when converting from x,y values to polar coord right?
@& Nothing is being converted to polar coordinates. This integral was in polar coordinates to start with. We could thinkg about what the corresponding integral would be in rectangular coordinates, but let's not get into that on this problem.*@
I mean that it is not necessary for our integral that is wrt `dr to have a mult of r in the integral, unless we are converting to polar coord and if not why does integral wrt `dr not have mult of r in the integral??????????
@& The area increment in polar coordinates is r dr dtheta. So this integral represents the area within the curve r = 1 + theta, for theta = 0 to pi. Just be sure you understand the graphs.*@
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Question: `q002. Use a double integral to find the area of the region described by 0 <= r <= 3 cos(5theta)
You should know from precalculus how to construct this graph, and how to test for various symmetries.
Between theta = 0 and theta = 2 pi, cos(5 theta) goes through 5 complete cycles from 1 to 0 to -1 and back to 0. Each cycles takes place in an angular distance of 2 pi / 5 (in degrees this is 72 degrees).
The figure below depicts the graph from theta = 0 to theta = 2 pi / 5.
The graph below adds the portion corresponding to theta = 2 pi / 5 to theta = 4 pi / 5, which appears in red.
The graph for theta = 4 pi / 5 to theta = 6 pi / 5 appears in green. Note that this graph traces over half the graph from theta = 0 to theta = 2 pi / 5.
The graph for theta = 6 pi / 5 to theta = 8 pi / 5 traces over more of the previous graphs, and adds nothing to the existing curve.
The graph for theta = 0 to theta = pi, in fact, completes the entire graph:
So the graph of r = 3 cos(5 theta) forms a sort of 5-leaved clover, and the domain 0 <= theta <= pi is sufficent to trace the region once.
A single 'leaf' is formed between theta = - pi/5 and theta = pi / 5. The area of the entire graph is therefore 5 times the area of this 'leaf'.
If this 'leaf' is partitioned into area increments, with typical area increment defined by the incremental angle `dTheta and increment `dr and containing sample point (r_hat, theta_hat), then the area of the increment is r `dTheta * `dr. The area does not depend on the sample point; if we were for example integrating to find mass then the mass density at the sample point would be important, and if we wished to find moment of inertia the density at the sample point and the r coordinate r_hat would be important.
The area of the 'leaf' is the total of the areas of the increments, and the resulting Riemann sum leads to the integral int( int( (1 * r dr), 0, 3 cos(theta)) dTheta, -pi/5, pi/5) In standard notation the integral is
The integral is easily evaluated. Using r^2 / 2 as our antiderivative of r, we find that the change in antiderivative for the inner integral is 9 cos^2(theta).
Integrating 9 cos^2(theta) we find that an antiderivative is 9 ( sin(theta) cos(theta) / 2 + theta / 2 ). It is possible using trigonometric formulas to get an exact expression for the values of this expression at theta = pi/5 and theta = - pi/5, but the approximation 4.967 is about as well as we can normally expect to do.
We can, however, get an exact solution. The antiderivative we obtained could easily be evaluated at theta = 0 and at theta = pi. This suggests just integrating over the entire domain.
The integral would be the same as before, except that limits on the theta integral would be 0 and pi rather than -pi/5 and pi/5. Our antiderivative would still be 9 ( sin(theta) cos(theta) / 2 + theta / 2 ). The change in the antiderivative from theta = 0 to theta = pi would be
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Your solution:
I did not look at solution
First we have a double integral with given boundries of 0<=r<=3cos(`theta), 0<=`theta<=2pi
Int(Int( 1 `dr,0,3cos(`theta)) `dtheta,0,2pi)
Int(1 `dr) = r,evaluated
3cos(`theta) - 0 = 3cos(`theta)
Int(3cos(`theta) `dtheta) = 3sin(`theta), evaluated
3sin(0) - 3sin(2`pi) = 0
??? When I was getting to the area for my solution I noticed your graph sort of(tried not to look), so for the integral to be equal to 0 does that mean the area is split evenly between the x axis. I mean there is just as much pos. area as area in the neg section. If so to keep from getting the wrong solution would we not just find area from 0 to `pi/2 and mult that value by 4 where `pi/2 is equal to (1/4) of `theta???????????
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Given Solution:
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Self-critique (if necessary):
I am going to have to go back and review the PreCal material. I don’t really have the time but I’m having a real hard time remembering everything that is going on with transformations in polar coord and effects on polar graphs.
The info on the graph was given in polar coord, so why did we have to put the extra r variable in the integral for `dr. I thought that we only had to mult fn by r when converting from (x,y) coord to (r, `theta) coord?
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@& The area increment in polar coordinates is r dr dtheta.
The region defined by the interval (r, r + `dr) and (theta, theta + `dTheta) is bounded by straight radial lines making angles theta and theta + `dTheta with the polar axis, and by short arcs of the circles of radii r and r + `dr (the circles being centered at the pole). The region is thus bounded by two straight line segments and two short arcs. The straight line segments have length `dr, while the arcs have lengths r dTheta and (r + `dr) dTheta.
r dTheta and (r + `dr) dTheta are nearly the same, since `dr is small, so we won't worry about the `dr dTheta. The two arcs are both about r dTheta in length. And for small dTheta they are both short enough to be practically straight.
The radial lines are perpendicular to the arcs, so for small `dr and small `dTheta the region bounded by the two radial lines and the two arcs is nearly rectangular.
The rectangle thus has dimensions very close to r `dTheta by `dr. That makes the area of the region r `dr `dTheta.
In the limit all approximations become exact, and the area of our region r `dTheta `dr becomes r dr dTheta.
So when we integrate to find area in polar coordinates, our area increment is r dr dTheta.*@
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Question: `q003. Use polar coordinates to find the double integral of f(x,y) = ln( a^2 + x^2 + y^2) over D with respect to A where D is the circular disk defined by x^2 + y^2 <= a^2 for some a > 0.
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Your solution:
We would need to use Cartesian coord for this problem, where r^2 = x^2 + y^2. To find the radius of the disk we are to calculate the area of we take what we are given wrt the dimensions of the disk which is
x^2 + y^2 <= a^2 for some a > 0, we can calculate the radius of this disk by
x^2 + y^2 <= a^2 r^2<=a^2, so we see that r<=a, so a is the radius of our disk.
Now we convert the fn which defines the area we are to calculate for with in this disk
ln( a^2 + x^2 + y^2) ln( a^2 + r^2), again doing so by the fact r^2 = x^2+y^2
Now we have our fn with which to integrate, the radius or r value we need to define part of the integration and all we need is `theta. `theta is easy to find because we are integrating over a circular disk all of which should have the same `theta which is 0<= `theta <= 2`pi
Remembering when converting from x,y coord to r,`theta coord to place and extra r value in the integral wrt `dr and recalling that we are integrating over entire disk so that our r value is 0 <= r <= a
We get
Int(Int(ln( a^2 + r^2)*r `dr ,0,a) `dtheta,0,2`pi)
Int(ln(a^2 + r^2)*r `dr), using u sub for a^2+r^2, du = 2r dr
Int( ln(u)*(1/2) `du) , using integration by parts u =ln(u),v’=1, du=1/u, v=u
Int( ln(u)*(1/2) `du) = (1/2)[u*ln(u) - Int(u*(1/u) du)] =(1/2)[ u*ln(u) - u]
(u(ln(u) - 1))/2 = ((a^2 + r^2)*(ln(a^2 + r^2) - 1))/2, evaluated
[((a^2 + a^2)*(ln(a^2 + a^2) - 1))/2] - [((a^2 + 0^2)*(ln(a^2 + 0^2) - 1))/2]
=(2a^2ln(2a^2) - 2a^2)- (a^2ln(a^2) - a^2))
Int((2a^2ln(2a^2) - 2a^2)- (a^2ln(a^2) - a^2))`dtheta,0,2`pi)
????I have gotten lost somewhere and can’t see my mistake going to look at solution?????
@& ln(a^2) = 2 ln(a). Once you see that you'll see that your integral is fine up to this point.*@
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Given Solution:
Using the conversions x = r cos(theta) and y = r sin(theta) we find that the disk x^2 + y^2 = a^2 is just r^2 cos^2(theta) + r^2 sin^2(theta) = a^2, or r = a (recall that sin^2(theta) + cos^2(theta) = 1).
Our function f(x, y) becomes f(r, theta) = ln(a^2 + r^2).
Our circular region is just 0 <= r <= a, 0 <= theta <= 2 pi.
Our integral is therefore int(int ( ln(a^2 + r^2) dr, 0, a), dTheta, 0, 2 pi).
The integral, and the inner integral, are shown below in standard form.
Our 'inner integral' is evaluated by the substitution u = a^2 + r^2, so that du = 2 r dr and our existing r dr is 1/2 du.
INT( ln(u) du / 2) = 1/2 (u ln(u) - u) + c, where c is an integration constant. Using 1/2 (u ln(u) - u) = 1/2 ( (a^2 + r^2) ln((a^2 + r^2) ) - (a^2 + r^2) ) as our antiderivative we find that from r = 0 to r = a, the antiderivative changes by a^2 ln(2 a) - a^2 / 2.
Our outer integral will therefore be
int( (a^2 ln(2 a) - a^2 / 2) dTheta, theta, 0, 2 pi) = (a^2 ln(2 a) - a^2 / 2) * 2 pi = 2 pi a^2 ln(2 a) - pi a^2.
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Self-critique (if necessary):
I’m not sure what you did when evaluating for the first integral, could you please explain even though I’m sure it simple algebra. I’m pretty sure that my mistake hides in there somewhere???
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Question: `q004. Find the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 5 and below by the parabola z = x^2 + y^2.
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Your solution:
Having a hard time seeing what I need to do to approach solution so I will look at solution provided for starting point then try and finish it out
Ok, so I’m still not noting everything perfectly but we have sphere centered at origin and parabola centered at origin and moving up in y direction. Rewriting our equations
x^2 + y^2 + z^2 = 5 x^2 + y^2 + z^2 - 5 = 0
z = x^2 + y^2 x^2 + y^2 -z
Subbing in r^2 = x^2 + y^2
r^2 + z^2 - 5 = 0
r^2 - z = 0
To find intersection we not z = r^2 from 2nd equation giving us
r^2 + (r^2)^2 - 5 = 0
= r^2 + r^4 - 5 = 0
To get in the appropriate form for quadratic formula we let b = r^2
b + b^2 - 5 = 0, resulting in b = (-1 +- `sqrt(21))/2, where -1 -`sqrt(21) is neg so it’s not a solution for r = `sqrt(b)
r = `sqrt((-1 - `sqrt(21))/2)
??? So why do we not have an second intersect, I am going to have to read further into solution to understand completely what’s going on. First time just read as little as possible to get an idea????
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Given Solution:
We must first find the curve of intersection between these two surfaces.
We could substitute z = x^2 + y^2 for z in the first equation, obtaining the equation (x^2 + y^2) ^ 2 + x^2 + y^2 = 20.
Expanding the square and substituting u = x^2 and v = y^2 we would get the equation u^2 + 2 u v + v^2 + u + v = 20.
We could try to complete the square but that u v term looks hopeless. It isn't really so, but there has to be an easier way to proceed.
We resort to a common shortcut. We note that x^2 + y^2 occurs in both equations. Furthermore, in polar coordinates x^2 + y^2 = r^2, so our equations can be written
r^2 + z^2 = 5 and
z = r^2.
This looks better than what we had before, so we proceed to eliminate z between the two equations.
Substituting r^2 for z in the first equation gives us
r^2 + r^4 = 5.
Letting u = r^2 our equation becomes
u^2 + 4 u - 5 = 0,
with solutions
u = 1, u = -4.
Since u = r^2, we have r = sqrt(1) = 1. (Note that u = 4 doesn't yield a solution since sqrt(-4) is not defined.)
Thus the curve of intersection lies above the circle r = 1 in the x y plane.
Since the origin is on the graph of the parabola, but lies below the graph of the sphere, it follows that the sphere lies above the parabola for all points within this circle.
On a typical area increment `dA defined by `dTheta and `dr, with sample point (r_hat, theta_hat):
• The region between the parabola and the sphere consists of a cylinder whose cross-sectional area is `dA, and whose sample-point altitude extends from the parabola to the sphere above the point (r_hat, theta_hat).
• The parabola lies at height z = r_hat^2 above the sample point, and the sphere at height sqrt( 20 - r^2 ).
• The cross-sectional area `dA is roughly a rectangle of dimensions r_hat * `dTheta by `dr, and the altitude is sqrt(20 - r_hat^2) - r_hat^2, so the volume of the increment is (sqrt(20 - r_hat^2) - r_hat^2) * r_hat * `dr * `dTheta.
• Summing over partitions of the intervals 0 <= r <= 1 and 0 <= theta <= 2 pi we obtain a Riemann sum, which approaches the integral
int (int( (sqrt(20 - r^2) - r^2) * r dr, 0, 1) dTheta, 0, 2 pi)
The inner integral is
int( (sqrt(20 - r^2) - r^2) * r dr, 0, 1) = int( sqrt(20 - r^2) * r dr, 0, 1) - int( r^3 dr, 0, 1) ,
which is fairly straightfoward (just let u = 20 - r^2 in the first, so that r dr = - 1/2 du, leaving int( -sqrt(u) / 2 du); and the integral of r^3 is very straightforward).
The result:
.
The definite integral simplifies to approximately 1.96.
There is no theta dependence, so the integral with respect to theta just multiplies this result by 2 pi, leading to a final result of approximately 12.3.
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Self-critique (if necessary):
I understand fairly well but can you help me understand what it takes to calculate the area between the two shapes, how you found these positions of the shapes. I know you explained but I’m not quite taking it all in. I’m pretty sure that I will never attempt to take another online math course, even though I feel I have the discipline and smart enough to make it work it takes so much out of you trying to teach myself some of these concepts. I had to do it to graduate but I don’t know what I was thinking.
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@& See my notes. You're doing OK here, asking the right quesitons, etc..
I clicked past the last problem accidentally and can't go back, but will post this again with notes on that problem. Also your query 20 and 21 documents for Mth 279.*@