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Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.$$f(x)=\frac{1}{x+1} ;[a, b]=[0,1]$$

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Calculus 1 / AB

Calculus 2 / BC

Chapter 5

INTEGRATION

Section 1

An Overview of the Area Problem

Functions

Limits

Differentiation

Integrals

Integration

Integration Techniques

Continuous Functions

Michael A.

May 9, 2020

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

03:14

Estimate the area between …

02:51

02:48

03:13

So this problem We're working with the function f of X is equal to one over X plus one working on the interval from 0 to 1, we're gonna be looking at two rectangles, five rectangles and 10 rectangles. The the more rectangles we use, the closer area will be to my final. To my more precise answer, we can use the sun because we're gonna have Teoh take each of the different pieces to find each of the rectangles were Take the some his end goes from 1 to 2 of 1/1 over and notice I've replaced the X with one over And cause this would be the the With of the rectangles some will be the height of the of the rectangles. Sorry. The with here is gonna be one over And okay, so I'm gonna take the some multiplied by the withs and we will get our final answer. When I plug this into my calculator, I'm gonna get approximately 0.5833 This will be an approximate area under the curve. We using two rectangles when I use five rectangles. I'm gonna slightly change the formal from 1 to 5. I noticed. The only thing that changes is the number rectangles. The width of the rectangles will also change will be 1/5 of the of the value compared to half of the value. So once I take the some, I'm gonna be multiplying by this value here. So that's going to give me 0.71 which is much closer to the area under the curve, but not quite there yet. When I repeat this process for 10. Yeah, and make my with 1/10 of a unit. I'm gonna end up with 0.7980 I can continue to go. Aiken 6100. The more rectangles they used, the closer my area gets to the actual here.

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