Find The Volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$

Question

Find the volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$.

$x$ and $y$ range from $0$ to $2$.

$$\int_0^2 \int_0^2 y-3 \,dy\,dx $$

is correct?

Answer 1

First if $0\le y\le2$ then we are going to need to integrate (3-y) or we will get a negative number.

To find the limits of y, I suggest you draw a line that is parallel to the y-axis. How would you represent the endpoints of this segment?

$x^2 + y^2 = 4\\ y^2 = 4-x^2\\ y = \sqrt{4-x^2}$

$\int_0^{2} \int_0^{\sqrt{4-x^2}} (3-y) dydx$

Now, if you want to convert to cylindrical, that is up to you, and probably not a bad idea.

Answer 2

Given that you are working with a cynlinder, it is easier to work with cylindrical coordinate.

$y=r\sin \theta.$

Also, notice that the hyperplane intersect the cylinder above $z=0$.

$$\int_0^{\pi/2} \int_0^2 (3-r\sin\theta) rdrd\theta$$