Arc length and surface area revisited, Mathematics

Assignment Help:

Arc Length and Surface Area Revisited

We won't be working any instances in this part.  This section is here exclusively for the aim of summarizing up all the arc length and surface area problems. The arc length and surface area has arisen several times and each time we got a new formula out of the mix.  Students frequently get a little overwhelmed along with all the formulas. Though, there really aren't as several formulas as it might seem at 1st glance.  There is precisely one arc length formula and exactly two surface area formulas.  These are as follow:

L = ∫ ds

S = ∫ 2Π y ds                           rotation about x - axis

S = ∫ 2Π x ds                           rotation about y - axis

The problems come up as we have quite a few ds's that we can utilize. Once again students frequently have trouble deciding which one to use.  The instances/problems generally suggest the correct one to use.  Now here is a total listing of all the ds's that we've seen and when they are employed.

If y =f (x), a < x < b then

ds = √ (1 + (dy/dx)2) dx

If x =h(y), c < y < d then

ds = √ (1 + (dx/dy)2) dy

If x =f (t), y = g (t), α < t < β then

ds = √ ((dx/dt)2 + (dy/dt)2) dt

If r = f (θ), α < θ < β then

ds = √ (r2 + (dr/dθ)2) dθ

Depending upon the type of the function we can speedily tell which ds to use. 

There is just only one other thing to worry about in terms of the surface area formula.The ds will make sure a new differential to the integral.  Before integrating ensure all the variables are in terms of this new differential.For instance if we have parametric equations we'll make use of the third ds and then we'll need to ensure and substitute for the x or y depending upon which axis we rotate regarding to obtain everything in terms of t.

Similarly, if we have a function in the form like x = h(y) then we'll make use of the second ds and if the rotation is regarding the y-axis we'll require to substitute for the x in the integral.Conversely if we rotate about the x-axis we won't require to do a substitution for the y.


Related Discussions:- Arc length and surface area revisited

Divides a given line-segment externally in the ratio of 1:2, Divides a give...

Divides a given line-segment externally in the ratio of 1:2 Construction: i )Draw BX making an actueangle at B. ii) Starting from B, mark 2 equal points on BX as shown in the f

Euler equations, Euler Equations - Series Solutions to Differential Equ...

Euler Equations - Series Solutions to Differential Equations In this section we require to look for solutions to, ax 2 y′′ + bxy′ + cy = 0 around x0  = 0. These ki

Evaluating functions, Next we have to talk about evaluating functions.  Eva...

Next we have to talk about evaluating functions.  Evaluating a function is in fact nothing more than asking what its value is for particular values of x. Another way of looking at

Characteristics and limitations of moving average, Characteristics and Limi...

Characteristics and Limitations of moving average Characteristics of moving average 1) The more the number of periods in the moving average, the greater the smoothing

What is the net area to be painted, An elevated cylindrical shaped water to...

An elevated cylindrical shaped water tower is in require of paint. If the radius of the tower is 10 ft and the tower is 40 ft tall, what is the net area to be painted? (π = 3.14)

What percent of the figure below is shaded, What percent of the figure belo...

What percent of the figure below is shaded? Break the rectangle into eighths as shown below. The shaded part is 6/8 or 3/4; 3/4 is 75%.

Calculus, I need help with my calculus

I need help with my calculus

Fermats theorem, Fermat's Theorem  If f(x) has a relative extrema at x...

Fermat's Theorem  If f(x) has a relative extrema at x = c and f′(c) exists then x = c is a critical point of f(x). Actually, this will be a critical point that f′(c) =0.

Determine does this point lie on the line, Does this Point Lie on The Line?...

Does this Point Lie on The Line? How do you know if a point lies on a given line? For example, does the point (1, 2) lie on the line 3x + y = 7? If you graph the line and the

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd