Find the volume bounded by the paraboloid $x^2y^2=az$, the cylinder $x^2y^2=2ay$ and the plane $z=0$ My work Changing to cylindrical coordinates4 EX 1 Sketch a graph of z = x2 y2 and x = y2 z2 5 A cylinder is the set of all points on lines parallel to l that intersect C where C is a plane curve and l is a line intersecting C, but not in the plane of C l 6 A Quadric Surface is a 3D surface whose equation is of the second degreeGraph x^2y^24x=0 x2 y2 − 4x = 0 x 2 y 2 4 x = 0 Complete the square for x2 −4x x 2 4 x Tap for more steps Use the form a x 2 b x c a x 2 b x c, to find the values of a a, b b, and c c a = 1, b = − 4, c = 0 a = 1, b = 4, c = 0 Consider the vertex form of a parabola a ( x d) 2 e a ( x d) 2 e
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Graph of cylinder x^2 y^2=4-Circular cylinder, x 2 y 2 =4;The graph shown in Figure 2 is an example of a cylinder in multivariable calculus It might seem strange to classify the graph in Figure 2 as a cylinder After all, when we say the word "cylinder", it is more likely that the image of a soup can comes into our head than the image shown in Figure 2
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Okay, so we have mathz = x^2 y^2/math describing the paraboloid and we have mathx^2 y^2 = 2y/math describing the cylinder That's how they look like together We want the equation describing the cylinder to be in its conventional formNow we draw the graph parametrically, as follows > cylinderplot ( r,theta,sqrt (16r^2),r=04,theta=02*Pi);X2 z2 1 Example 3612 Reduce the equation to one of the standard forms, classify the surface, and sketch it 4y2 z2 x16y 4z =0 To solve this, we will have to complete the square The first step is to organize the equation by variable and factor out coecients of the highest degree term 4(y2 24y)z 4z x=0
The x2 by 4, then the xvalues will only lie between −1/2 and 1/2 and thus the graph would be an ellipsoid with a smaller radii in the xdirection Example 23 Describe and sketch the quadric surface z = x2 y2 For any fixed value of z = k > 0, in the plane z = k, the trace (or crosssection) is a circle of radius k There are no solutions(e) Below is the graph of z = x2 y2 On the graph of the surface, sketch the traces that you found in parts (a) and (c) For problems 1213, nd an equation of the trace of the surface in the indicated plane Describe the graph of the trace 12 Surface 8x 2 y z2 = 9;Example 015 % The Viviani's Curve is the intersection of sphere x^2 y^2 z^2 = 4*a^2 %and cylinder (xa)^2 y^2 =a^2 %This script uses parametric equations for the Viviani's Curve,
Weekly Subscription $249 USD per week until cancelled Monthly Subscription $799 USD per month until cancelled Annual Subscription $3499 USD per year until cancelledSo the intersection of the cylinder x 2 y = 9 and the surface z = xy can be represented by (3cos(t),3sin(t),9cos(t)sin(t)) 3( pts) Find a vector function that represents the curve of intersection of the cone z = p x 2y and the plane z = x2 Solution Both equations (z = pAnswer (1 of 3) It's the equation of sphere The general equation of sphere looks like (xx_0)^2(yy_0)^2(zz_0)^2=a^2 Where (x_0,y_0,z_0) is the centre of the circle and a is the radious of the circle It's graph looks like Credits This 3D Graph is created @ code graphing calculator
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The problem is try to sketch the hand The curve of intersection of the circular Salinger X squared plus y squared is secret war, and there's a parabolic sine you Z is equal to X squared Then find a parametric equations for this curve and the youth fifty equations and a computer to graph the craft First, let's sketch a hand It's a curse First John graph After circular assigned er that wouldGraph the paraboloid z = 4x2 y2 z = 4 x 2 y 2 and the parabolic cylinder y= x2 y = x 2 Find the equation of the intersectionMA 351 Fall 07 Exam #3 Review Solutions 4 6 Evaluate ZZ D x p y2 x2dA, D= f(x;y)j0 y 1;0 x yg ZZ D x p y2 x2dA= Z 1 0 Z y 0 x(y2 x2)12 dxdy (use usub on y2 x2) Z 1 0 1 3 (y2 x2)3=2 x=y x=0 dy= Z 1 0 1 3 (y2 y2)3=2 1 3 (y2)3=2 dy Z 1 0 1 12 y3 dy= 1 12 y4 1 0 = 1 12 7 Find the volume of the solid under the paraboloid z= x 2 y 4 and the planes x= 0, y= 0, z= 0 and x y= 1
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Graph the parent quadratic (y = x^2) by creating a table of values using select x values The graph of this parent quadratic is called a parabolaNOTE AnyCircular cylinder, x 2 z 2 =4;The graph of a function f(x;y) = 8 x2 y) So, one surface we could use is the part of the surface So, one surface we could use is the part of the surface z= 8 x 2 yinside the cylinder x 2 y
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21 Match The Equation X 2 4y 2 9z 2 1 With Its Graph Labeled I Viii Toughstem
No, see, we have to make for all four of these funds And so here the table Now this is the table And now we want X and y values So here is X and Y put x zero Put a X equal to the wise minus three Put X equals to one So he had put excellent Why equal So this will minus toe Good X equal to minus one See the explanantion This is the equation of a circle with its centre at the origin Think of the axis as the sides of a triangle with the Hypotenuse being the line from the centre to the point on the circle By using Pythagoras you would end up with the equation given where the 4 is in fact r^2 To obtain the plot points manipulate the equation as below Given" "x^2y^2=r^2" ">" "x^2y^2 =4If one of the variables x, y or z is missing from the equation of a surface, then the surface is a cylinder Note When you are dealing with surfaces, it is important to recognize that an equation like x2 y2 = 1 represents a cylinder and not a circle The trace of the cylinder x 2 y = 1 in the xyplane is the circle with equations x2 y2
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(tcost)2 (tsint)2 = t2 cos2 tt2 sin2 t= t2(cos2 tsin2 t) = t2 It lies on z 2 = x 2 y 2 Find a vector function that represents the curve of intersection of the paraboloid Show Solution Okay, since we are looking for the portion of the plane that lies in front of the y z y z plane we are going to need to write the equation of the surface in the form x = g ( y, z) x = g ( y, z) This is easy enough to do x = 1 − y − z x = 1 − y − z Next, we need to determine just what D D isSinusoidal cylinder, y = sin(x) Sinusoidal cylinder, z = sin(x) From economics we have the important concept of a CobbDouglas production function, the simplest example of which is f(x,y) = In economic terms, the function relates productivity to labor and capital The graph of this function for 0 < x < 2 and 0 < y < 2
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Answer (1 of 11) There's a simple answer, if you don't wish to think — you can find it in all the other answers given But I'll assume you'd like to understand what's happening here I tutor fifth and sixthgrade students and this is exactly how I'd describe it to them The graph of x^2 y^2We have given Why equals toe X cubed minus three? Probably you can recognize it as the equation of a circle with radius r = 1 and center at the origin, (0,0) The general equation of the circle of radius r and center at (h,k) is (x −h)2 (y −k)2 = r2 Answer link
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Since the surface is in the form x = f ( y, z) x = f ( y, z) we can quickly write down a set of parametric equations as follows, x = 5 y 2 2 z 2 − 10 y = y z = z x = 5 y 2 2 z 2 − 10 y = y z = z The last two equations are just there to acknowledge that we can choose y y and z z to be anything we want them to beThe graph of equation is a cylinder with radius centered on the yaxis In this case, the equation contains all three variables and so none of the variables can vary arbitrarily The easiest way to visualize this surface is to use a computer graphing utility (see the following figure) Convert to vertex form ( y = a (x b)^2 c where (b, c) is the vertex ) y = x^2 4x y = (x 2)^2 4 so the vertex is at ( 2, 4) answer
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Answer to Find the area of the part of the surface z= 4xy that lies within the cylinder x^2 y^2 less than or equal to16 r(s,t)= (s cos(t),= 2t Find the equation of the plane through the point P and orthogonal to the lineGet stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!
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Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack ExchangeNow suppose that the cylinders and sphere are sliced by a plane that is parallel to the previous one but that shaves off only a small portion of each cylinder (have a look at the picture on the left) This will produce parallel tracks on each cylinder, which intersect as before to form a square cross section of the volume common to both cylindersAlgebra Graph x^2 (y2)^2=4 x2 (y − 2)2 = 4 x 2 ( y 2) 2 = 4 This is the form of a circle Use this form to determine the center and radius of the circle (x−h)2 (y−k)2 = r2 ( x h) 2 ( y k) 2 = r 2 Match the values in this circle to those of the standard form The variable r r represents the radius of the circle, h h represents the xoffset from the origin, and k k represents the y
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For example, if the equation is $(x1)^2 (y2)^2 = 4$, then one of the points on the cylinder is (1,0,0), but so is (1,0,1) and (1,0,1) and (1,0,5) and (1,0, 7) and so on This means that the cylinder goes on forever both up and downCalculus questions and answers The graph of x2 2 = 1 is an ordinary right circular cylinder Indicate axis of the cylinder lies along and the radius of cylinder Use , between axis and radius Given the point P (4, 5, 1) and the line L x= 3 2t;Popular Problems Algebra Graph x^2y^2=4 x2 − y2 = 4 x 2 y 2 = 4 Find the standard form of the hyperbola Tap for more steps Divide each term by 4 4 to make the right side equal to one x 2 4 − y 2 4 = 4 4 x 2 4 y 2 4 = 4 4 Simplify each term in
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Level surfaces For a function $w=f(x,\,y,\,z) \, U \,\subseteq\, {\mathbb R}^3 \to {\mathbb R}$ the level surface of value $c$ is the surface $S$ in $U \subseteqNext, let us draw the cylinder x^2 y^2 = 2 In this cylinder, the radius r is always 2 We let theta vary from 0 to 2*Pi as usual, and let z range from 0 to 4 to match the the height of the sphere that we just drewProblem 64 Hard Difficulty (a) The plane y z = 3 intersects the cylinder x 2 y 2 = 5 in an ellipse Find parametric equations for the tangent line to this ellipse at the point ( 1, 2, 1) (b) Graph the cylinder, the plane, and the tangent line on the same screen
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Traces are useful in sketching cylindrical surfaces For a cylinder in three dimensions, though, only one set of traces is useful Notice, in Figure 280, that the trace of the graph of z = sin x z = sin x in the xzplane is useful in constructing the graphThe trace in the xyplane, though, is just a series of parallel lines, and the trace in the yzplane is simply one linePrecalculus Graph y^2=4x^2 y2 = 4 − x2 y 2 = 4 x 2 Move −x2 x 2 to the left side of the equation because it contains a variable y2 x2 = 4 y 2 x 2 = 4 This is the form of a circle Use this form to determine the center and radius of the circle (x−h)2 (y−k)2 = r2 ( x h) 2 ( y k) 2 = r 2 Match the values in this circle to those of the standard formOkay, are copies of the same ellipse in the plane Z equals K So it follows that the surface for X squared plus y scrape was for is an elliptic cylinder which has rulings parallel to the Z axis With this description, I can catch the graph of this surface We have our X, y and Z axes, and first I'll draw the ellipse in the X Y plane
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Plane z = 1 The trace in the z = 1 plane is the ellipse x2 y2 8 = 1X 2 y dxdy where Dis the region inside the curve r= 2 and outside the curve r= 4cos in the rst quadrant 2Find the volume of the solid underParabolic cylinder, y 2 = z Parabolic cylinder, z 2 = x;
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Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack Exchange It is an equation of a circle with centre at (0,0) and radius 9 The equation is of the form of a circle with center at origin, as in the general form of a quadratic equation ax^22hxyby^22fx2gyc=0, while coefficients of x^2 and y^2 are equal (ie a=b), f, g, h are all zeros In fact equation x^2y^2=81 graph{x^2y^2=81 , , 10, 10} can be written as (x0)^2(y0)^2=9^2Graph y=x^24 y = x2 − 4 y = x 2 4 Find the properties of the given parabola Tap for more steps Rewrite the equation in vertex form Tap for more steps Complete the square for x 2 − 4 x 2 4 Tap for more steps Use the form a x 2 b x c a x 2 b x
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