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Evaluate the integral by interpreting it in terms of areas.

$ \displaystyle \int^0_{-3} \bigl( 1 + \sqrt{9 - x^2} \bigr) \, dx $

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02:31

Frank Lin

00:54

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Maya R.

November 26, 2021

Very well explained, thank you.

Missouri State University

Baylor University

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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in order to evaluate this under girl here, we could first graft dysfunction. And to do so, we could actually just graph this nine minus sex weird, which is a circle. And then it shifted up one. And therefore it has a radius three, but then shifted up one unit and therefore it has a Y intercept up for So basically, from here below, we just have a big rectangle and that indicates how much it shifted. Now we want to find the area from negative 3 to 0. So we want to find this area as well as the area of the semi circle. So let's just start with the rectangle. Well, that's just the height of one width of three and therefore that has in the area of three. They're positive. And then what about thesis Urkal? The area for a circle is pi r squared. So it's just write that x squared our radius. And this case is three so deep I times three square. But then we're only interested in 1/4 of it the circle. So it's over four or another way to rewrite that is nine pie sports. That is the area of this top region here. So adding those two together, we would get that the Senate girl evaluates to using geometry three plus nine high over four. And we could just go ahead and leave this because this is our exact answer.

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