Download the App!
Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.
Question
Answered step-by-step
The region bounded by the curve $ y = \frac{1}{(1 + e^{-x})} $, the x and y-axes, and the line $ x = 10 $ is rotated about the x-axis. Use Simpson's Rule with $ n = 10 $ to estimate the volume of the resulting solid.
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by Foster Wisusik
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 7
Approximate Integration
Integration Techniques
Missouri State University
Oregon State University
Harvey Mudd College
Boston College
Lectures
01:53
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
27:53
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
03:22
The region bounded by the …
05:05
The region bounded by $y=\…
03:16
The region bounded by $y=e…
03:57
The region under the curve…
09:02
The region bounded by $y=1…
Okay, so this question wants us to estimate the volume of the salad formed by revolving this bounded region around the X axis. So if it's bounded by the red curve, the green line, the blue line and the purple line well, a region is just this thing here and well, what is that? Well, that's just the area underneath the graph of the red curve between zero and 10. So f of X equals one over one plus eat of the minus X, and our radius of our show is just equal to F of X. So our volume we're using the disc method is just pi times the integral from 0 to 10 of f of X squared DX, which in this case is pie times the Inter girlfriend 0 to 10 of one over one plus e to the minus x squared D X. So now we have our expression for the volume and we need to use Simpson's rule to approximate it. So it says use and equals time. So Delta X is we're ending at 10 starting at zero with 10 to sub intervals. So Delta X is one. So that means that s a turn is approximately equal to the volume and Essam 10 equals well, Delta X is one so we get one over three times F zero squared plus four f of one squared plus two f of two squared plus all the way up to F of 10 squared with a factor of pi Tek Don because pie comes along with the volume integral. So if we evaluate this, we see that the volume there's approximately equal to 28 and again First we found our region to set up our volume integral right here and then to find the integral. We just used Simpsons Rule being extra careful to use our factor of pie in front and two square every one of our F values like it is in the integral.
View More Answers From This Book
Find Another Textbook
01:55
Which statement describes the term economics? A. It is the study of usin…
02:46
Dropping less than two inches per mile after emerging from the mountains, a …
01:59
If you roll two fair dice (one black die and one white die), in how many way…
01:13
The area of an extra large circular pizza from Gambino's Pizzeria is 48…
00:55
Two radical expressions are called like terms if they have the same degree a…
02:13
The scale for a map is 20 miles = 1/2 inch. The distance between two towns o…
01:06
In the fire Academy we had a ratio of 7 to 5 passing grades to failing. How …
00:45
What is the multiplicative rate of change for the exponential function f(x) …
02:41
A horse-drawn carriage tour company has found that the number of people that…
02:56
The blue whale is the largest animal living on the earth. The average blue w…