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Evaluate the integral by interpreting it in terms of areas.

$ \displaystyle \int^9_0 \biggl( \frac{1}{3}x - 2 \biggr) \, dx $

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01:42

Frank Lin

00:46

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Madalyn J.

January 1, 2022

Evaluate the integral by interpreting it in terms of areas. 8 ?7 (1 ? x) dx

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

05:53

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.

40:35

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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evaluate the integral.…

evaluate this integral using area. We could first graph the function the Inter grand there in green. And then we wanna look at the area under the curve from negative or from 0 to 9. So obviously we have a lot more negative area than we have. Positive area. So there's going to be a negative and a girl here, Let's go ahead and find out exactly what the area of this triangle is. Andi, the base is going to be six because that's where it crosses the X axis and move that the the height is to so six times 2 12 divided by 21 half base times height would get off an area here of positive six and then we want toe. Analyze the area here. But sorry, negative sex is what I meant. And then we wanna analyze the area for this triangle up top. Here it has a base of three and a height of one. So three times one divided by two would be three halves. So this has an area here of three halves, positive or negative or positive 1.5. So if we combine these now, we have negative sex below the X axis, so it's negative plus 1.5, and that would end up giving us the integral evaluates to negative 4.5. Or, of course, negative. Nine out is the same thing there as well, because that would end up evaluating to negative four and a half. So by splitting these up into triangles, using the area equals one half base times height and then identifying anything below the X axis is negative above. The X axis is positive, then that could be a way to combine these to find the area.

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