Find the volume of the solid in the firsr quadrant bounded by the coordinate planes, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=3$

Question

Find the volume of the solid in the first quadrant bounded by the coordinate planes, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=3$.

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If we draw the graph, then the integral will be calculated should be

$$ \int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}} (3-y) \: dy dx $$

with $3-y = z = f(x,y)$.

The boundary $ \sqrt{4-x^{2}} $ is from the cylinder. Is this correct? Thanks.

Answer 1

Yes, that is correct. And if you are going to evaluate the integral, I suggest polar coordinates.