Find the length and the cuur vature of the Smooth eurve rCt) = (snt)i t(5 cost); + (sint)K from o to api . where t goes

Q: given ∫∫s (x2z) dS and S is the part in z = sqrt(x2+y2), which lies in x2+y2 = 1 calculate the…

A: we have to evaluate ∬Sx2zdS Now, we have z=x2+y2 ⇒z2=x2+y2∂z∂x=xz and…

Q: Given y dx – xy? dy where D is described by æ? + y? <1 -

A:

Q: Use the arc length formula to show that the length of a linear function f(x) = mx + c on the…

A: Let f(x) be a function over the interval (a,b), Then the arc length of f(x) from the point (a,f(a))…

Q: Find the area of the surface swept out by revolving the para- metric curve x = 5 cost + 5t sin t y =…

A: Area under curve.

Q: Show that at any point (x,y) on the curve y=bex/a ,the length of the subtangent cosgant and the…

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Q: Use the arc length formula to find the length of the linex = t, y = t, for 0 ≤ t ≤ 1.

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Q: Find the length of the equiangular spiral r = e° for 0 <o< T. L =

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Q: For the curve y = x^2 + 1 between the points (1,2) ??? (3,10) : a. find the Arc Length b. and find…

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Q: What does the arc length formula give for the length of the line y = x between x = 0 and x = a,…

A: line y = x between x = 0 and x = a

Q: Let f(x) = x³/2. Determine the length of the arc swept over x E la, b).

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Q: The work done W,to stretch a spring from its natural length to an extension of 6 meters Is given 6.…

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Q: |Find the average distance from the origin over all the points in the unit circular disk 2² +y? <1.…

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Q: Use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces…

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Q: Show that at any point (x,y) on the curve y=bex/a , the length of the subtangent is constant and the…

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Q: Find the mass of a thin wire shaped in the form of a helix, r(t) = ( 3 cos t, 3 sin t, 4t) with 0 <…

A: The curve is given by  rt=⟨3cost,3sint,4t⟩ , 0≤t≤3π2                   (i) Then,…

Q: x = r², y = (0 <1 < 1)

A: For x=x(t), y=y(t), a≤t≤b Arc length formula is L=∫abdxdt2+dydt2 dt

Q: Set up an intergral for the length of the curve, 04X y%3DZE* sin cx), O4 XE

A: Explanation of the answer is as follows

Q: Find the arclength of r(t)= on the interval from t equals zero tot equals 2 times pi.

A: Given:  r(t)=&lt;5cot(t), 3 sin(t), 4 sin(t)&gt;To find:  Arc length form t=0 to t=2π

Q: At which of the points below is the tangent plane to the surface z - 2y + z?+ yz = 2 vertical?…

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Q: 4. Given YZ tangent to oJ at point Y and m<WYZ = 104, %3D what is the measure of WXY?

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Q: Find the moment of inertia, with respect to the line x=1, of the area enclosed by the loop of the…

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Q: 4. Find the length of the arc C = {(x, y): y = V16 – x²?,0 < x < 3}.

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Q: Find the speed of the cycloid c(t) = (4t − 4 sin t, 4 − 4 cost) atpoints where the tangent line is…

A: Given: c(t)=4t-4sin(t), 4-4cos(t)

Q: Find the length of the subtangent and subnormal of a point of the curve y=b sin (x/a)

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Q: H use spherical Cingidinates to find the velumi of the oliet inside xtty42²=9 and outside z=ty" and…

A: Given sphere, x2+y2+z2=9 outside z=x2+y2   and above xy plane. We use spherical coordinates to find…

Q: Let f(x) = x3/2. Determine the length of the arc swept over r E (a, b).

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Q: Find the arc length of that portion of the circular helix x = cos t, y = sint, z = t from t = 0to t…

A: Find the arc length of that portion of the circular helix x = cos t, y = sint, z = t from t = 0to t…

Q: Given YZ tangent to OJ at point Y, and mzWYZ = 104°, what is mW XY ? W mWXY

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Q: The lateral (side) surface area of a cone of height h and base radius r should be Ar v? +h?, the…

A: Consider the provided question, Given the line segment, y=rhx,  0⩽x⩽h Differentiate both side of…

Q: Find T N ¢B for the carve at the indlicated pont Carレe posnt r(t) = at

A: Given

Q: 4. Use the arc length formula to show that the length of a linear function f(x) = mx +c on the…

A:

Q: Use Simpson’s Rule with n = 10 to approximate the length of arc of r(t) = t i + t2j + t3k from the…

A: Given,

Q: A circle of radius a in the xy plane, centred at the c is described by T = ¢ +acos (t) i +asin (t)…

A: In this question, we need to change the orientation of the path.

Q: Determine the resulting surface area when the curve y = x is rotated about the y-axis in the…

A: Note:- Since you have asked multiple question As per our policy, we will solve the first question…

Q: 2. At what point on z = a? + y? is the tangent plane parallel to 3.r – 4y + z = 7 ?

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Q: Approximate the arc length of the curve y = x over the interval [1, 2] using the trapezoidal rule…

A: Given y=14x4 in [1,2] , 7 sub intervals  y'=14×4x3 = x3 Area length = ∫ab1+y'2dx…

Q: Consider the pavametenzed cunve given by X=cos, y= sin (t), 0ste2 TT a) Sketch the curve b) set up…

A: SOLUTION-

Q: C is the part of the circle = 1 in the first quadrant, find the following line integral with respect…

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Q: Find the curvature K of the curve, where s is the arc length parameter. V2 r(s) : K =

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Q: Find the value of a such that the arc length of the catenary y = cosh x for −a ≤ x ≤ a equals 10.

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Q: Find the lengths of tangent, subtangent, normal and subnormal to y=4ax at ( at,2at).

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Q: What does the arc length formula give for the lengthof the line r(t) = ⟨2t, t, -2t⟩, for 0 ≤ t ≤ 3?

A: Given:          r(t)=⟨2t,t,−2t⟩  for 0≤t≤3.

Q: A figure on d coordinate plane is dilated using the rule (x, y).

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Q: Find the equetion of the targent pane to tre grven Surfface at the Specified point Z=XSin(x+y) at…

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Q: Find the average x-coordinate of a point on the half-torus with the following parametrization (drawn…

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Q: Find the area of the surface obtained by rotating thecircle x2 + y2 = r about the line y =r.

A: We have to find the area of the surface obtained by rotating the circle x2 + y2 = r about the line y…

Q: Sketch the grapm of the pare cone (6) and Sketeh TE) and valve of f. quaph N't) at the %3D

A: For a vector valued function r→t=a(t)i→+b(t)j→, its unit tangent vector T→t and unit normal vector…

Q: Find an arc length parametrization of the line y = 8x + 7. r;(s) II

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Q: By revolving the semicircle x = r cos t, y = r sin t. (0 ≤ t ≤ π) about the x-axis, show that the…

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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.