5. Do the following tasks using Mathematica. i) Find the length L of the curve y = 1 In a, 1<¤< 2 Use any one of the following formulas ,dy, dx' 2 L = 1+ :) dr 2 da. dy 1+(- L = lip,

Q: Given the double intergral 3 S.³ To √²+y² Evaluate the above dy dx intergral using cartesian…

A: some of the formula of integration . ∫1x2+a2dx=ln|x+x2+a2|+C∫xndx=xn+1n+1+C

Q: Find an equation of the tangent to the curve given by = evi y = t – In(?) at the point corresponding…

A:

Q: 3 The arc length of the portion of the curve y =x² with 0<x<4 is

A:

Q: 3. What is the length of the curve r = cos3 (-) where 0 < 0 <- 4

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Q: Find the length of each of the following curves: %3D %3D x (ey + e) from y = 0 to y = 1

A: The length of a curve means the arc of the length of a curve. The arc of the length of a curve in…

Q: 2. Find the length of the curve 0<t< 2n. r(t) e'i + e' sin 2tj + e' cos 2tk,

A: From the given problem :   rt=eti+etsin2tj+etcos2tk   So,   x=et, y=etsin2t , z=etcos2t .   As we…

Q: Find the arclength of the curve a = 8 cos(3t), y = 8 sin(3t) with 0 < t<. 9

A:

Q: 3. Find the length of the curve y = In (cos x), 0 <x <

A: we have to find length of the curve

Q: Compute the arclength of the following curves. (y-1)³ =x², for 0 <x< 5 а.

A: We have to compute the arc length for the function y-13=x2, for 0≤x≤5.

Q: Consider the curve defined by xy + y2 = 21 Determine dy/dx for this curve at the point (4, 3)

A:

Q: Evaluate Iz = y", 2ry ) · d7 where C3 is the arc of the curve x = 3 – 2y that extends from (7, –2)…

A: we have to evaluate the following

Q: Let C be the curve defined by 7(t) = (2+2 sin(t), 3 – 2 cos(t)) for 0 <t< L (e + 4z) dz + (2zye" +…

A:

Q: (a) (., Consider the curve defined by ry = In(3x + sin(y)). Use implicit differenti- ation to find…

A: Consider the given equation of the curve, xy=ln3x+siny Use the implicit differentiation,…

Q: 3. Find the exact arc length of the curve I = y+}y2 from y = 1 to y = 4.

A:

Q: Find the arclength of the curve x= 4cos(6t), y= 4sin(6t) with 0 less than or equal to t less than or…

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Q: Find the arclength of the curve y = * - In x if < y s-1. Do not simplify your answer for this…

A: Given: y=18x2−lnx18≤y≤e28−1

Q: Find 4 and 4y axz dx y=te* dy- -(t-1)et y t is Fov which values the curve con cave upwavd. U BRANDS

A:

Q: Find the tangent line(s) to the parametric curve given by a = t5 – 4t3 y = t? at (0, 4).

A: Find the tangent line(s) to the parametric curve given by x = t^5-4t^3 , y = t^2 at (0,4).

Q: Find the arc length (8) of the curve 7 (t) = ( – 2e*, – 4e*, – 2e*) fo %3D c - - -

A: Explanation and solutions is given below....

Q: If lx + my = 1 touches the curve (ax)" + (by)" = 1, show that %3D )+) = 1.

A:

Q: 3 Consider cune detined by y cos (T2) =UY +y+2 2 ) Verify point (2,-2) belongs to this curve, fond…

A:

Q: 1. Find the exact arc length of the curve 24.xy = y* + 48 %3D from y = 2 to Y = 4.

A: To find the exact arc length of the curve,24xy =y4+48from y=2 to y=4.

Q: Compute the arc length of the indicated portion of the curve: 7(1) = (7 +2r²)î + (21² – 4)ĵ + (7 –…

A:

Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

A:

Q: 3 The arc length of the portion of the curve y=x 2 with 0<x<4 is

A: I have append the formula used to find arc length in the solution part.

Q: 2. Determine the arc length of the given curves a) y =+ 1<y< 2 3

A: s=∫y1y2 1+x'2 dyy'=3y23-14y2y'=y2-14y2s=∫121+y2-14y22dys=∫121+y4+116y4-2×y×14y dys=∫12y4+116y4+12…

Q: 7. Find the arc length of the following plane curve: x = e' (sint + cos t), y = e' (cost – sin t),…

A:

Q: Can there be a curve satisfying the following conditions? d2y/dx2 is everywhere equal to zero and,…

A: To check: The curve satisfying the given condition or not.Explanation:

Q: dx L% = 1+ dy dy

A: Given equation of curve; y  = (x/2)3/2 , x = 0 to x = 2

Q: Starting from the point (2, –1, 2), reparametrize the curve r(t) = (2 – 2t, -1+ 2t, 2 – 3t) in terms…

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Q: - Find the aera under the curve y = cosx Form x = 0 to x = -fx sec² (ax² )dx a#0

A:

Q: Answer! 6. Compute yz dz + zzdy+ zydz along the curve (cos t, sin t, tan t). 0sts*

A: line integrals

Q: Calculate the length of the curve γ (t) = (cos(2t), sin(2t)) t ∈ [0, π].

A:

Q: 1) The Arc Length of the curve f(x) = coshx, 0≤x≤ In3 is: d. 312 12 3 C. b. 2²/2 0.33 lorod by…

A:

Q: 2. Let y be the curve in R3 given by t (t, 2 cos t, 2 sin t) for 0 <t < 47. Compute ds

A:

Q: Consider the parametric curve: X y = ln(-t) Which integral gives the arc length of the curve over…

A: We have to find the length of curve.

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: 3) Find the exact length of the curve. y = In(cos r), 0 < x<

A:

Q: Consider a curve parameterized by: c(t)=(sin^3(t),cos(t)) Compute dy/dx at the point ((sqrt(2)/4),…

A: Differentiate the given function as follows.

Q: Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 In(cos(t)) k, osts n/4

A: Given:

Q: 3 dz where C, is the top half of circle with radius (b) Compute the upper bound of 2,

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Q: 1 Find the arc length of the curve y = + 4 from (0, 4) to (1, ).

A:

Q: Find the arclength of the curve æ = 5 cos(4t), y = 5 sin(4t) with 0 < t s 4

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Q: Consider the parametric curve: X = = 7et y = 3t5 Which integral gives the arc length of the curve…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

Q: Find (2x + xy)ds, where C is the arc of the curve y = sin (x) for 0 ≤ x ≤ π

A: The given problem is to evaluate the given line integral of the given function over the given curves…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.