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Numerade Educator

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Problem 37 Hard Difficulty

Evaluate the integral by interpreting it in terms of areas.

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

Answer

$$3+\frac{9}{4} \pi$$

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Video Transcript

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.