Find the curve length of the given functions in the given interval a) x=e'cost, y = e'sent0stsb) xt, y= t - vī , , 0sts1

Answered: a) x=e'cost, y = e'sent 0sts b) xt, y=…

Math

Calculus

a) x=e'cost, y = e'sent 0sts b) xt, y= t - vī , , 0sts1

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.3: Cylinders And Cones

Problem 6E: Suppose that r=12 cm and h=15 cm in the right circular cylinder. Find the exact and approximate a...

See similar textbooks

Related questions

Q: 4. Find equations of the tangent plane and the normal line to the surface yz = arctan(2x+z) at the…

Q: Find the area of the segment cut off from the curve y = sin x, 0 ≤ x ≤ ¹ by the line y = 1/√2. 3r 4…

Q: Find the between the two curves. area 2. X=ay, a'y=x 3

A: The area between the curves can be calculated by integration. The formula used for the integration…

Q: Find a point on the surface x+4y--z² =1 where the tangent plane is parallel to the plane x+4y- z =…

A: we need to find gradient of given surface to know about direction of normal vector to tangent plane

Q: Q1) Find equation of line normal to plane 3x + 2y - z = 5 and pass through Origin point. 30 mark

Q: What is the equation of the tangent plane to the surface y = eª cos (z) at the point (1, e, 0)?

Q: Exercise (c) Show that y = Vx has a vertical tangent line at (0, 0). ...

A: first we will find derivative of function. So, derivative will be slope of tangent. for vertical…

Q: Find the tangent to y = cotx at x = - T O A. y = - 2x + 2 π +1 +1 OB. y=2x-- OC. y=2x+1 OD. y=-2x +…

Q: Sketch the recion enclosed by the given curves. Decide whether to integrate with respect to rwr.Daw…

A: Draw all the regions first then decide which region satisfy all the curves

Q: rea under the curve y = 2x – x² from x = 1 to x = 2 with n = 4 %3D tonglon

Q: 1 dt 7+t+3t2 y =

Q: Find equation of line normal to plane 3x + 2y- z = 5 and pass through Drigin point.

Q: Find the equation for the tangent plane and the normal line at the point Po(1,1,3) on the surface…

A: Given problem:- Find the equation for the tangent plane and the normal line at the point Po(1,1,3)…

Q: 6) What is the tangent plane of f (x, y) = 5x² – 2xy? + 3xy + 4x at point (xo, Yo) = (1, –2) ?

A: The question is solved in step. 2

Q: 2. Find the slope of the tangent in the positive x-direction and y-direction to the surface z = 2xy…

Q: 5. Find the length of the curve given by x = cos t +t sin t, y = sin t – t cost, -T <t <

Q: Зп 2. Find the exact length of curve y = ln (cosx), * < x < n. 4

Q: A Bifolium is the family of curves given by (x +y)* = kx²y, where k is a real number constant. %3D…

Q: Find the equation of the tangent plane to the surface 3 – 12y – sin(r*:) = 0 at the point (2, 1,0).…

A: To find gradient of the function.

Q: Find a curve through the origin in the xy- plane whose length is given by 1/2 L = [; (1+ *)"* dx 4

A: The arc length, L of a curve y=f(x) in the interval a,b is calculated using the formula below.…

Q: 2. Find the length of arc of the curve x = + from the point where y = 1 to the point where y = 2. yª…

A: Explanation of the answer is as follows

Q: Find the equation on the plane tangent to the following surfaces at a given point: z=exy; (2,0,1)…

A: The equation on the plane tangent to the surface z = f(x, y) at a given point (a, b, c) is given by:…

Q: Use the Trapezium Rule with four strips to estimate the value of: dz. (In z)² Give your answer…

A: Use the formula of the Trapezium Rule to solve the integral. This rule is used to solve the integral…

Q: 5. Find the length of the curve given by y= Vt* – 1 dt and 1<x<2.

Q: Evaluate the line mtegral over the curve S(-x + y +z) dx + (x+y - z) dy + (x-y +z) dz where the line…

Q: Justify why the level curves very close to the origin (0,0) for the function f(z, y) = (1-z)(1 – )…

A: As per the instruction, we will be solving the first question. For (1), We are given the function as…

Q: Compute ∫C(2x+√y)ds, where Cis the curve from (0,0) to (2,4) along y=x^2.

Q: Find equation of line normal to plane 3x + 2y – z = 5 and pass througl Origin point.

Q: Find equations of the normal plane and tangent line to the curve of intersection of the surfaces…

Q: Find the tangent plane of the function f(x, y) = √2x² + 2y² at (1,1). Is it parallel to 2- the plane…

Q: Find equations for the a) tangent line and b) normal plane to the curve 1 x=t- cost, y = 3+ sin 2t,…

A: x=t-cost, y=3+sin2t, z=1+cos3t We know that rt=xt, yt, zt dxdt=1+sint, dydt=2cos2t, dzdt=-3sin3t At…

Q: "my A curve y given by the range of z: [a, b] → Cis said to be : 1) Smooth if ...... 2) Piecewise…

A: Explanation of the answer is as follows

Q: What is the moment of inertia of this area about the line y = -10 ?

Q: 9. Find an equation for the tangent plane and the normal line to the surface xy + yz + zx = 5 at the…

A: f(x,y,z)=xy+yz+zx-5=0, at (1,2,1)now we differentiate above equation with respect to x,y,z…

Q: Find the equation for (a) the tangent plane and (b) the normal line at the point Po(4,0,4) on the…

A: The above problem is solved.

Q: A thin plate of constant density d = 1 occupies the region enclosed by the curve y = 36/(2x + 3) and…

Q: Q1) Find equation of line normal to plane 3x + 2y - z = 5 and pass through Origin point.

Q: Calculate the curve integral ∫c ( 3x2 - 3yz + 2xz )dx + ( 3y2 - 3xz +z2 )dy + ( 3z2 - 3xy + x2 +2yz…

A: Using green's theorem to find the line integral.

Q: Find the equation for (a) the tangent plane and (b) the normal line at the point Po(8,0,8) on the…

Q: Find a curve through the point (1, 9 whose length on the interval [1, 2] is given by i VI+ 25z!0dz-…

A: Explanation: Given a curve, ∫121+25x10dx Function f(x) has an arc length on the interval [a,b] is…

Q: 4. Find the equation of the tangent plane to the level surface x4 + yz – yz² + 5x = 0 at 3

Q: A plane area is bounded in the first quadrant by the curves y² = -16 (x-4), the line 8x-15y = 0, and…

A: Given the curve is y2=-16x-4 and the line 8x-15y=0 and x-axis. We have to find the moment of inertia…

Q: Find a curve in the xy-plane passing through the origin andwhose slope at the point (x, y) is x + y.

A: we have to find a curve in the xy-plane passing through the origin and whose slope at the point (x,…

Q: Find equations for the tangent plane and the normal line at point Po (Xo Yo.Zo) (5,2 on the surface…

A: We know that the equation of the tangent plane is: nxx-x0+nyy-y0+nzz-z0=0 Given:…

Q: Q2: Determine the tangent plane and normal line to the surface x^2+2xy-y^2+z^3=7 at point (1,-1,3)

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: 1)Determine the equation of the tangent line to the intersection of the surfaces x²-y=0 and y+z²=16…

Q: Give arc-lengtu porametriz atian of the line which an Hhe tangent planes of and (1,2,9). is Hhe…

Q: The orea bounded by the curve y=- ond the line xty=3 is equal to ?

A: The given curve is , y=2x...........(1) and the line x+y=3...........(2) we have to find area…

Concept explainers

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.

Question

Find the curve length of the given functions in the given interval

a) x=e'cost, y = e'sent
0sts
b) xt, y= t - vī , , 0sts1

Expert Solution

Step 1

Find dx/dt and dy/dt using product rule.

Then find (dx/dt)^2 and (dy/dt)^2.

y=e'sint
dy
= (e' )'sint+e'(sint)
x-e'cost
dx
=(e')'cost+e'(cost)
dt
dt
dy
=e'sint+e'cost
dt
dx
=e'cost-e'sint
dt
dy
(e'sint+e'cost)
dt
dx
(e'cost-e'sint)
dt
2.
dy
e2 cost 2e2*cost sint+e2 sin't
dt
dx
e2 cost -2e2cost sint+e2 sin't
dt

Step 2

Plug them in the formula of curve length and simplify.

Answer(a): sqrt(2)[e^pi - 1]

cost -2e2 cost sint+e2 sin't +e2 cos't +2e2*cost sint+e2 sin't dt
L=
= V2e2 cos2t +2e2 sin2t dt
= 2e2(cos't +sin°t ) dt
L
dt
L=
L=\2e'dt
L=v2[e-e°]
L=V2 [e-1]

Step 3

Find dx/dt and dy/dt

And then find (dx/dt)^2 and (dy/dt)^2

y=t-Vt
x=t+ t
1
dy
=l-
dt
dx
=1+
dt
2T
2Vt
dy
1
1-
2Vt
(
dx
1+
dt
2t
dt
1
dy
dx
1
=1+
1
=1.
dt
4t
4t
dt

Step by step

Solved in 6 steps with 6 images

Knowledge Booster

$Application of Integration$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.