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Use a graph to give a rough estimate of the area …

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Problem 51 Medium Difficulty

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sin x $, $ 0 \le x \le \pi $


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Frank Lin

00:41

Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Related Topics

Integrals

Integration

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Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

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Kristen Karbon

University of Michigan - Ann Arbor

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Integrals - Intro

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

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40:35

Area Under Curves - Overview

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Video Transcript

for this problem. We are examining the equation y equals sine of X, and we're looking at values of X from zero to pi inclusive. Now we want to find the area under this curve. We're going to do it two ways. First, we're going to actually look at the graph and kind of get an estimate of what we expect to see. Then we're going to use calculus to find the exact value. So first you can use a calculator graphing calculator, a graphic application on your computer, whatever you have that you're comfortable with. For this example, I am using the desk Most graphing application on my computer. I have programmed in Why equal sign effects, which is our equation from zero to pi, As you can see for decimus in order to give that range, I just put it inside curly brackets. Your application, if you use something different than the nomenclature, might be a little different. But you can see I have one hump going from the X axis back down to the X axis again. It's above the X axis. So I'm expecting a positive number. Well, if I look at this, if I imagine taking the piece The excess from 2 to 3, that little triangle piece that almost fits and makes a rectangle that would have a base of two and a height of one. I mean, it's not perfect, but it's awfully close. So I'm going to estimate Let me just write this down here. My estimate is that we're going to be around two units for our for our answer just from looking at that graph. Now, let's find our exact answer using calculus. So to find the area under the curve, we're going to take the integral. I'm evaluating this integral from X equaling zero to pi. So those are my limits of integration of sine x dx? Well, what is the integral of sine? The integral of sine is negative co sign of X okay. And I want to evaluate this from X equals zero two X equals pi. So let's put these in. We always start with our upper limit first. So if I sake excess pie co sign of pie is negative one. I'm taking the opposite of that. So that's a one. Now I subtract the value at the lower limit. X equaling zero co sign of zero is one. Take the opposite of that. It's going to be negative one. So one minus negative one. Well, that's adding. So actually, my answer is to my estimate, was exactly right.

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Video Thumbnail

40:35

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In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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