Surface area- applications of integrals, Mathematics

Assignment Help:

Surface Area- Applications of integrals

In this part we are going to look again at solids of revolution. We very firstly looked at them back in Calculus I while we found the volume of the solid of revolution. In this part we wish to find the surface area of this region.

Thus, for the purposes of the derivation of the formula, let us look at rotating the continuous function

y = f (x) in the interval [a, b]

about the x-axis. Below is an outline (sketch) of a function and the solid of revolution we obtain by rotating the function about the x-axis.

2405_Surface Area- Applications of integrals 5.png

We can obtain a formula for the surface area much more like we derived the formula for arc length. We'll initiate by dividing the integral into n equal subintervals of width Πx. On each subinterval we will estimated the function with a straight line that agrees along with the function at the endpoints of the each interval. Below is a sketch (figure) of that for our representative function using n=4.

165_Surface Area- Applications of integrals 4.png

 

Here, rotate the approximations about the x-axis and we get the subsequent solid.

1958_Surface Area- Applications of integrals 3.png

The approximation on every interval provides a distinct portion of the solid and to make this clear every portion is colored differently. Each of these portions are termed as frustums and we know how to find out the surface area of frustums. The surface area of a frustum is illustrated by,

A = 2πrl

r = ½ (r1 + r2)

r1 = radius of right end

r2 = radius of left end

the length of the slant of the frustum.

For the frustum on the interval [xi-1, x1] we contain,

R1 = f(xi)

R2 = f(xi-1)

l = |Pi-1 Pi| (length of the line segment connecting pi and pi-1)

and we know from the preceding section that,

|Pi-1 Pi| = √ 1 + [f' (xi*)]2 Πx

where xi* is some point in,

[Xi-1, Xi]

Previous to writing down the formula for the surface area we are going to presume that Πx is "small" and since f(x) is continuous we can then assume that,

F (xi) » f (xi*) and f (xi-1) » f (xi*)

Thus, the surface area of the frustum on the interval [Xi-1, Xi] is approximately,

Ai = aΠ (f (xi) + f (xi-1) / 2) |pi-1 pi |

» 2Π f (xi*) √ 1+ [f'(xi*)]Πx

After that the surface area of the whole solid is approximately,

2080_Surface Area- Applications of integrals 2.png

and we can obtain the exact surface area by taking the limit as n goes to infinity.

2422_Surface Area- Applications of integrals 1.png

If we wish to we could as well derive a similar formula for rotating x = h(y) on [c,d] about the y-axis. This would provide the following formula.

S = ∫dc 2Π h (y) √ (1+ [h' (y)]2) dy

Though, these are not the "standard" formulas. Note: The roots in both of these formulas are nothing much more than the two ds's we employed in the previous section.

As well, we will replace f(x) with y and h(y) with x. By doing this gives the following two formulas for the surface area.


Related Discussions:- Surface area- applications of integrals

Quantitative techniques, mentioning the type of business you could start an...

mentioning the type of business you could start and the location of your business, use the steps of quantitative methods for decision making narrating them one by one in the applic

Linear equation, The ratio between the length and breadth of a rectangular ...

The ratio between the length and breadth of a rectangular field is 11:7. The cost of fencing it is Rs. 75,000. Find the dimensions of the field

Linda bought 35 yards of fencing how much did she spend, Linda bought 35 ya...

Linda bought 35 yards of fencing at $4.88 a yard. How much did she spend? To multiply decimals, multiply generally, count the number of decimal places in the problem, then us

To calculate volume of cylinder which formula is used, Mimi is filling a te...

Mimi is filling a tennis ball can along with water. She wants to know the volume of the cylinder shaped can. Which formula will she use? The volume of a cylinder is π times the

The mean value theorem for integrals of even and odd , The Mean Value Theor...

The Mean Value Theorem for Integrals If  f (x ) is a continuous function on [a,b] then there is a number c in [a,b] such as,                                    ∫ b a f ( x

Illustrate median with example, Q. Illustrate Median with example? Ans...

Q. Illustrate Median with example? Ans. The median of a data set is the middle value (or the average of the two middle terms if there are an even number of data values) wh

Expressions, how do you solve expressions

how do you solve expressions

Example for comparison test for improper integrals, Example for Comparison ...

Example for Comparison Test for Improper Integrals Example:  Find out if the following integral is convergent or divergent. ∫ ∞ 2 (cos 2 x) / x 2 (dx) Solution

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