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Sketch the area represented by the given definite integral.$$\int_{-3}^{3} \sqrt{9-x^{2}} d x$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

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University of Nottingham

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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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Sketch the region whose ar…

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Sketch the integrand of th…

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Sketch the region of integ…

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Use the graph of the integ…

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sketch the region of integ…

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Okay, So when you look at this problem and were just asked to sketch the area, um, and we could find the answer. It's not that difficult of this problem. What I would do, first of all, is just established that these are your balance. X equals negative three and X equals positive three. I guess I should clarify that I'm looking at these numbers so those would be vertical lines. Except I'm not actually gonna draw the vertical line, because if I were to plug in negative three in for X Um you know, So I called this f A bex, I guess. And I try to figure out of negative three. Well, negative three square to be positive. 99 minus 90 in the square. 20 is zero. So instead of drawing a vertical line, I know it's going to hit the graph. I could do the same thing with three. That's equals three. Um 99990 in the square. 20 is still zero. So maybe just try one more value. I'll do F zero. Well, nine miles zero is zero in the square to nine is three. So that tells me is that the graph? Look, something like this, and because I have experienced with this I know that this is a semicircle. Um, and we only want the positive version of it. So it's a semicircle that looks something like this. And hopefully you have some sort of familiarity with semi circles and equations of circles. So the area that we're examining is just this piece in here. Um, and if you wanted to, you could actually find the area of it, because it's area of a circle Is pi r squared, but divided by two. Well, since the radius is 39 pi over two or nine has pie for the answer for the area. Anyway, this is the graph.

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