y = cos(x), 0

Q: 4. Use the Arc Length in Parametric Form formula to find the arc length of the curve with the given…

A: Given parametric equations, x= 3sint ,   y=3cost,     0≤t≤π3

Q: (1) Evaluate the line integral 4xy dx + (2x² – 3xy)dy if the curve C consists of the line segment…

A:

Q: 1. Find an equation for the tangent plane to the surface x cos z + yře*z =4 at Po(3, –1, 0)

A:

Q: 4. Find the length of arc of the curve x3 + y3 = 1 in the first quadrant from the point where x = to…

A: If a function f(x) is defined over an interval [a,b], then the formula for arc length is: Formulae…

Q: 2. Write an integral expression to represent the length of the path described by the parametric…

A:

Q: 5. Eliminate the parameter to find a Cartesian equation of the following curves. et (a) x = vt+1, y…

A: Eliminate parameter t.

Q: 5. Eliminate the parameter to find the equation of the ellips Sx(t) = 4 cos(t) + 1 ly(t) = 3 sin(t)…

A:

Q: 1-4 Find the area of the surface generated by revolving the given curve about the x-axis. . 1. y =…

A: Surface area is 7✓55 π

Q: 1-4 Find the area of the surface generated by revolving the given curve about the x-axis. 2. y = ,…

A:

Q: 4. (i) Use the parametric equations of an ellipse, x = a cos (t) , y = b sin (t), 0 < t 2m, to find…

A:

Q: 9) Find the area of the surface generated by rotating about the y-axis y = x' between (0,0) and…

A:

Q: 4. Find the equation of the tangent plane to the surface Z = = 5tan(x) – sin(3y) at the point (xoyo)…

A:

Q: 1. Give an equation of the tangent plane to the surface z= In(4y – x) at the point (3,1,0).

A:

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

A:

Q: (7.5.0)- Find an equation of the tangent plane to the surface of z = sin(x – 2y) at the point %3D

A: Given F(x, y, z)=z-sin(x-2y) ∂∂xF(x, y, z)=∂∂x(z-sin(x-2y))=-cos(x-2y)∂∂yF(x, y,…

Q: 4. Consider the surface S parametrized by R(u, v) = (uv, sin u, cos v). Find an equation for %3D the…

A: The given curve S is parametrized by: R→(u,v)=uv,sin u,cos v Now, to find the equation of the…

Q: |3| 7/7 2) Find the arc length of y = ln (sec x) from x = 0 to r 11

A:

Q: 3/ 10. Find the arc length of the curve y = x/2 +4 from x = 4to x = 9.

A:

Q: 1. a) Find the length of the curve: y=+[1,2]. b) Find the area of the surface obtained by rotating…

A:

Q: Compute the surface area of a surface generated by revolving the curve c(1) = (6 sin²(1), 6 cos² (1)…

A:

Q: 4 Find the arc length of the curve the interval 2≤+ ≤ 5 3 given parametrically by x= 3 + ² y = + ²,²…

A: To find out the arc length of the curve.

Q: An equation for the tangent plane to the surface z + 3 = xy cos (2) at the point (3, 1, 0) is:

A:

Q: 4. Find the radius and centre of curvature of the curve x = (1 - at) cos t + a sin t, y = (1 – at)…

A:

Q: Sketch the regions enclosed by the given curves. y = 2 cos(5x), y 2 sin(10x), x = 0, x = 1/10 2.0…

A:

Q: 1. Find the arc length of the curve x = r=(y²+ 2)2 from y-0 to y-1

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: 9.40. Show that the 0-parameter curves on the surface x = (r cos 0)e¡ + (r sin 0)eg + f(0)eg are…

A: We have to show that the θ-parameter curves on the surface x are parallel.

Q: 1 4) Find the exact area of the surface obtained by rotating the curve x = + on the interval 6 2y…

A:

Q: 1. Eliminate the parameter to find a Cartesian equation of the curve given parametrically by (x(t),…

A: Recall: sin2t+cos2t=1

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. х 3D sin?t, у— cos t yA

A:

Q: 3 4. Find the arc length of the graph of the function f(x) =;(x² + 1)ī over the interval [0, 1]. =…

A: y =f(x) =23(x2+1)32f(x)=23(x2+1)32f'(x)=23*32(x2+1)12(2x)f'(x)=(2x)(x2+1)12 The Formula for arc…

Q: 6. Consider the same curve from the last problem, C parametrized by T(t) = (2 cos t, –1,2 sin t)…

A:

Q: 5. Consider the two curves r= / cos(0) and r= V2 cos(0), -T/2<0ST/2,whose polar plo 0.6 0.4 0.2 0.2…

A: Here, the given curves are: r=cos θ &amp; r=2cos θ

Q: 3 6. Find the arc length of f(x) = G (x2 + 2)z on the interval [0,9]

A:

Q: 1. Find an equation of the tangent plane to the surface of z = sin(x – 2y) at the point (7.5.0).

A: First partially differentiate the function with respect to x,y and z and then proceed as shown…

Q: 4. Find the equation of the tangent plane to the surface ey + In(xz) – evz = 0 at the point (1,0,…

A: Given:

Q: Page 9 9. The curve y = x2 with 0Sx<1 is rotated about the y-axis. Find the exact area of the…

A:

Q: 1(a) Find the arc length of the curve 2x = 3y2/3 over the interval from y=0 to y=1  1(b) Find the…

A: (a) The arc length of the curve 2x=3y23 over the interval y=0 to y=1 can be calculated by using…

Q: 1-4 Find the area of the surface generated by revolving the given curve about the x-axis. 2. y = ,…

A:

Q: 1. Find the equation of the tangent plane to the surface cosh [exp (x²y)] = Z y? at the point where…

A: We will find out the required equation of tangent plane.

Q: 4. Find the arc length of the curve x=y+y from y-1 to y-4

A:

Q: #4

A: Find the tangent plane to the given surface at the specified point

Q: 1 Find the area of the surface generated by revolving the curve x = cos (3t), y 3+sin (3t) on 0sts…

A:

Q: 7. Find an equation of the tangent plane to the given surface at the specified point. cos(x –…

A:

Q: b-Two curves x = 0 – sin 0, y = 1 - cos 0 are rotated around the y-axis. Calculate the area of the…

A:

Q: Q1. Find the length of the of the curve y = In(cos x) from x = 0 to x = . %3D

A: Explanation of the solution is given below...

Q: 1.) Use the arc length formula to find the length of the curve y = 3x + 2 on the interval 0 ≤ x ≤ 3.

A: Given data  y=3x+2 Interval ÷ 0≤x≤3 Formula for arc length is given by  L=∫ab1+dydx2dx

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.