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Sketch the region of integration and evaluate the following integrals as they are written.$$\int_{0}^{2} \int_{0}^{4-y^{2}} y d x d y$$

$$2(2)^{2}-\frac{(2)^{4}}{4}=4$$

Calculus 3

Chapter 14

Multiple Integration

Section 2

Double Integrals over General Regions

Multiple Integrals

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the saintly want us to sketch a region? Uh, integration. Well, let's see. We're gonna be computing. Do you want from 0 to 2? Why is that Call and see what are actually computing as Let's see. Alright. Starts at zero. On the top curve is a bookings home for wine to get a square. Actually, is that necessary? We already know that it's going backward. We're starting at four. The shift hope for And then go Teoh. There we go. So we're competing this region over here when I go from 0 to 2 toe where it intersects the, um Lanxess. So let's see, this will be equal to the interrupt from zero to see ah x one evaluated up for minus y squared zero b y. But zero we'll just get a zero. So we're only gonna be concerned with evaluating four minus y squared. But they don't just give us for away, man. That's why Cube do you want. And now we could take the anteroom of science so we'll be evaluating up the endpoints fundamental calculus. That's two y squared minus y for a before evaluated at 20 Well, we really that zero will just get zero. So let's evaluate at two eight minus. Let's the square to 16/4. It just gives us a for in the end.

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