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# Curves

### Stokes' Theorem : Circulation of a Field Around a Curve

Please see the attached file for the fully formatted problems.

### Contact of Sphere and a Sphere Surface

See the attached file. A spacer defines the air-gap distance between a lens and a flat surface. The corner of the spacer can be modeled as a circle with radius, r1. First a circular lens is used, and the "sag," or amount that the lens bulges out on the axis, defines the air gap distance by: sag = R - (R2 - (chord)2) ½ In th

### Contact Point of Two Circles

If two circles with radii r and R contact at a point above a straight axis, what is the height, &#916;y, of contact (as a function of the curvature of the circles) above the axis. Please see the attached file for the fully formatted problem.

### Arc Length Intervals

Find the arc length: x^2 = (y-1)^3 on [0,8] 0<=x<=8

### Sketching three-dimensional curves.

Prob: Sketch the region enclosed between 2 - z = x2 + y2 and z2 = x2 + y2. Describe their curve of intersection

### Finding the Curve's Horizontal and Vertical Tangents

Find the points where the curve r=3sin2θhas horizontal and vertical tangents.

### Cycloids : Area and Arc Length

See attached file for full problem description.

### Equation of a Parabola

(3) A wagon is pulled a distance of 100m along a horizontal path by a constant force of 50N. The handle of the wagon is held at an angle of 30 degree above the horizontal. How much work is done? (4) Find an equation of a parabola that has curvature 4 at the origin.

### Curvature Functions

Find k(t) for x = e^(3t) , y = e^(-t) .

### Arc Length of a Parametric Curve

Find the arc length of the parametric curve x = 3 cos t, y = 3 sin t, z = 4t; 0 <= t <= pi .

### Finding Curvature of a Function

Find the curvature 1. y =x^3 2. y =cosx

### Find the length of the curve

Find the length of the curve. See attached file for full problem description.

### Arc Length

Find the arc length of the graph f (x)= (2/3)*(x-6)^(3/2) on the interval [6,12]

### Area of Region Bounded By Curves

Find the area of the region bounded by the graphs of y=x^2 - 4x and y=x -4. keywords: finding, find, calculate, calculating, determine, determining, verify, verifying

### Area under a Curve

Calculate the area under the curve y=1/(x^2) above the x-axis on the interval [1, positive infinity]. keywords: finding, find, calculate, calculating, determine, determining, verify, verifying

### Finding Curvature

Compute the curvature k(t) of the curve r(t) = 2t i + 4sint j +4cost k

### Locus of a point (determining the equation of a curve)

A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve.

### Area under the curve

Find the area of the surface obtained when the graph of y=x^2 , 0<= x <= 1, is rotated around the y axis.

### Eliminating the Parameter and Sketching a Curve

A.) Eliminate the parameter to find a Cartesian equation of the curve b.) Sketch the curve and indicate with an arrow the direction in which the cuve is traced as the parameter increases x= 4 cos t, y= 5 sin t, -pi/2 (</=) t (</=) pi/2.

### Higher-Order System : Unit-step Response Curve

Consider a higher order system defined by (see attached file for equation): a) Using MATLAB, plot the unit-step response curve of this system. b) Obtain the rise time, peak time, maximum overshoot, and settling time of the system. Please type it in Microsoft Word using Equation Editor. It makes it easier to understand.

### Curvature

I am asked to find the curvature at the point where t=1 of the curve given by: vector r(t) = t i + 1/2 t^2 j + 1/3t^3 k How do I do this problem and what is the final answer?

### Regions Bounded by Curves and Finding the Centroid

Please see the attached file for the fully formatted problems.

### Arc length : Find the length of a Curve (9 Problems)

Please see the attached file for the fully formatted problems.

### Curvature of a Curve in Space

The curvature of a curve in space r(t) is given by k(t) = | r'(t) à? r''(t) | / | r'(t) |^3 . Consider now the curve r(u) = r(sigma(u)), given by the reparametrization t = sigma(u) of the initial curve. Show that the curvature k of the curve r is given by k(u) = k(sigma(u)), where k is the curvature of the initia

### Arc Length of a Curve on an Interval

Find the arc length of the curve y=1/3(x^2+2)^(3/2) on the interval [0,1].

### Calculating Volumes of Bounded Regions (Curve, Origin and Interval)

1. Sketch the region bounded between the given curves and then find the area of each region a) {see attachment} b) Find the area of the region than contains the origin and is bounded by the line {see attachment} 2. Sketch the given region and then find the volume of the solid whose base is the given region and which has the

### Limits : Parametric Curves

Two equations (see attachment) with limits (dealing with parametric curves.

### Curve Sketching Step-by-Step

I need help using the guidelines of curve sketching to sketch y=(((x^2)-1)^2)^1/3.

### Question about Curve Sketching Step-by-Step

Please help using the guidelines of curve sketching to sketch y=(x^2)/((x^2)+3). The steps seem more complicated than they should be to me, and I can't seem to get anything remotely looking like a correct answer.

### Maximum and Minimum Values (Looking at Curves and Gradients)

For each of the following functions, find the maximum and minimum values of the function on the circular disk: x2 + y2 [less than or equal to] 1. Do this by looking at the level curves and gradients. *(For functions please see attachment)