Consider a unit sphere centred at the origin. Find an arc length parametrised geodesic connecting the points (0,0, 1) and (0,0, –1). Is the solution to this problem unique?

Q: Find the arc length of the curve C from the origin to the point (1/2, 1/6, 1) , where C' is the…

A: Given equations are z2=2x and 3y=xz. Now multiplying z2=2x with z:  z3=2xzz3=6y

Q: Show that the ellipse x = a cos t, y = b sin t, a 7 b 7 0, has its largest curvature on its major…

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Q: Use the parametric equations of an ellipse, x = 5 cos (0), y = 4 sin(0), 0≤ 0 ≤ 2π, to find the area…

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Q: Show that the parametrization of the ellipse by the angle 0 is ab cos e Va? sin? e + b2 cos2 0 ab…

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Q: Find the arc length parametrization of the helix r(t) = (cos 4t,sin 4t, 3t)

A: The arc length  of a differentiable vector valued function r(t) is given by the equation;…

Q: Determine the arc length parameterization of the circle x^2+y^2 =a^2 with orientation…

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Q: Use the parametric equations of an ellipse, æ = 2 cos(0), y = 5 sin(0), 0 < 0 < 27, to find the area…

A: Given :  x=2cos(θ), y=5sin(θ),  0≤θ≤2π The area under the curve can be written as:…

Q: Use the parametric equations of an ellipse, x = 3 cos (0), y = 4 sin(0), 0≤ 0 ≤ 2T, to find the area…

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Q: Chapter 12, Section 12.5, Question 013 Find the curvature and the radius of curvature at the stated…

A: The given vector-valued function is rt=6cost i+6sint j+7t k at t=π2 to find the curvature and the…

Q: 3) Find the measure of the arc length of AB in the your work on a separate sheet of paper 9 in. A…

A: If you  like the solution then please give it a thumps up... The Answer is:                 arc AB…

Q: Find the arc length of p = 21, 0 = In t, =, 1si5. V5 4V5 2V5 3/5

A: There are two ways to spherical polar coordinates, first confirm the notation uses... Although i…

Q: Find an arc length parametrization of r(t) = (et sin t,et cost,et)

A: Given that, rt=etsint, etcost, et       ...(1)

Q: Find an arc length parametrization of the circle in the plane z = 11 with radius 4 and center (10,…

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Q: Calculate the arc length of the entire cardioid r=a(1-cos).

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Q: Show that the parabolas $r=c /(1+\cos \theta)$ and $r=d /(1-\cos \theta)$ intersect at right angles.

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Q: A flashlight shape like a paraboloid has its light source at the focus located 1.5cm from the base…

A: Sol:     given:  4 flashlight shape like parabolaod                  and its light source of the…

Q: A beam whose centre of gravity divides it into two portions, a and b. is placed insıde a smooth…

A: I am attaching image so that you understand each and every step.

Q: Find the arc length form (2,1) to (4,4) of the curve x = 2/y.

A: The formula for arc length is L=∫abf'x2+1dy.

Q: Describe the projections of the circle r(t) =〈sin t, 0, 4 + cost〉onto the coordinate planes.

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Q: (a) exactly one point P in R³. Do NOT put one sphere inside the other one. Draw a picture of two…

A: a) The objective is to draw two spheres of different sizes tangent to each other at exactly one…

Q: Find the arc length of the graph of r(t) = ev2li+e¯v2lj+2tk, 0<t < v2 In 9 NOTE: Enter the exact…

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Q: In the figure PQSO, is a trapezium in which PQ || OS, ZPOS = 135° and ZOSQ = 90° points P, Q and R…

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Q: Al 21. What type of three-dimensional object forms when a semicircle is rotated around the a-axis?…

A: Clearly we can say that, when it rotated about x-axis then it forms a SPHERE.

Q: Use the parametric equations of an ellipse, x = 5 cos(0), y = 4 sin(0), 0 < 0< 2n, to find the area…

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Q: Calculate the distance on the ellipsoid if the distance on the secant is 18267.46

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Q: Write the equation for the sphere of radius 1 centered at the prigin in the each of the following…

A: The general equation of a sphere in cartesian coordinates is: x-a2+y-b2+z-c2 = r2 Here, r is the…

Q: Find parametric equations for the circle x2 + y2 = a2, using as parameter the arc length s measured…

A: The given equation of circle is,

Q: dV , where V is the unit sphere.

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Q: Find the EXCESS (E) and AREA of a spherical triangle whose radius is 12 m and angles are A=30°;…

A: Given : In the spherical triangle, ∠A=30°∠B=100°∠C=80° The radius of the sphere is 12 m We have to…

Q: A plane travels from point A on a bearing 150° to a point B 10 km from A as illustrated above. How…

A: We are given the following information. We need to find AC.

Q: . Find a parametrization for the circle (x – 2)² + y² = 1 starting at (1, 0) and moving clockwise…

A: Consider the equation of circle

Q: Find the arc length of the cardioid r = 2 – 2sin©.

A: Given that: r=2-2sinθ Differentiating with respect to theta,…

Q: Use the parametric equations of an ellipse, D=5 cos(@), y - 4 sin(6), 0 O 27, to find the area that…

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Q: Use the parametric equations of an ellipse, x = a cos(0), y = b sin(0), o sos 2r, to find the area…

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Q: ) convert the eylindrieal pt (3, 7%,-2) to rectomgular ) conrert the nectamgular ) to cylindrical

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Q: A satellite orbits the Earth with an elliptical orbit modeled by the maximum distance between the…

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Q: 4.2. Calculate the arc length of the cardioid given by r = a(1 – cos 0) -

A: The cardioid is symmetrical in nature and ranges from θ=0 to θ=π as a half figure. We can also…

Q: Give a parametric description for a sphere with radius a, includingthe intervals for the parameters.

A: We will use the spherical coordinates for parametrization. Where ρ= radius= a And,

Q: 1. Use the parametric equations of an ellipse, x = a cos 0, y = b sin 0, 0 < 0< 27, to find the area…

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Q: Let C be the curve represented by R(t) = (cos 2t, 2√3t, 5 — sin 2t). Find the coordinates of point Q…

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Q: Which of the following geometric solid would result after rotating a circle around its diameter? *…

A: When you will rotate a circle about it's diameter , you will obtain a sphere .

Q: 3. Givea parametrie description fora sphere with radis, ineluding the intervats for the peremeter

A: Given : radius = a

Q: Show that 8r units is the length, L, of one arch of the cycloid described parametrically by: x (0) =…

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Q: Which of the following is an arc-length parametrization of the circle in the plane z=10 with radius…

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Q: A hole is drilled through the center of a sphere x² + y² + z? = R,², leaving a ring-shape solid. If…

A: given sphere x2+y2+z2=R2s and a drill of cylinderical shape x2+y2=R2d use polar coordinate to…

Q: Use the parametric equations of an ellipse, x a cos(0), y = b sin(0), 0 seS 2n, to find the area…

A: Given that the parametric equation of an ellipse is  x= a cos(θ) , y=b sin(θ) , 0≤ θ ≤ 2π . Here we…

Q: what is the laplace equation for a sphere

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Q: Find the exact arc length of the curve on the given interval. Parametric Equations Interval x = 4…

A: Given: x=4cosθ, y=4sinθ; θ∈0,π for finding arc lengthL, first we convert given equation in…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.