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The first two $\iint$ represent your region of integration here you are using $\displaystyle \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} $ this is a very well known region, it's circular region.

In method $(2)$ When you intersect two cylinders, you get a square region as in the link.

The also note that the height function of this intersection of cylinders are different on different regions, as one cylinder's height is taller than other on some region. On region $(a)$ cylinder $(2)$ is taller than cylinder $(1)$ therefore cylinder $(1)$ acts are bounds for floor and ceiling.

The first two $\iint$ represent your region of integration here you are using $\displaystyle \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} $ this is a very well known region, it's circular region.

In method $(2)$ When you intersect two cylinders, you get a square region as in the link.

The also note that the height function of this intersection of cylinders are different on different regions, as one cylinder's height is taller than other on some region. On region $(a)$ cylinder $(2)$ is taller than cylinder $(1)$ therefore cylinder $(1)$ acts are bounds for floor and ceiling.

In method $(1)$ you take a circular region, and use another cylinder for bounds.

The first two $\iint$ represent your region of integration here you are using $\displaystyle \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} $ this is a very well known region, it's circular region. When you intersect two cylinders, you get a square region as in the link.

The also note that the height function of this intersection of cylinders are different on different regions, as one cylinder's height is taller than other on some region. On region $(a)$ cylinder $(2)$ is taller than cylinder $(1)$ therefore cylinder $(1)$ acts are bounds for floor and ceiling.

And I prefer the method $(2)$.

The first two $\iint$ represent your region of integration here you are using $\displaystyle \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} $ this is a very well known region, it's circular region.

In method $(2)$ When you intersect two cylinders, you get a square region as in the link.

The also note that the height function of this intersection of cylinders are different on different regions, as one cylinder's height is taller than other on some region. On region $(a)$ cylinder $(2)$ is taller than cylinder $(1)$ therefore cylinder $(1)$ acts are bounds for floor and ceiling.

The first to $\iint$ represent your region of integration here you are using $\displaystyle \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} $ this is a very well known region, it's circle. When you intersect two cylinders, you get a square region as in the link.

The also note that the height function of this intersection of cylinders are different on different regions, as one cylinder's height is taller than other on some region.

The first two $\iint$ represent your region of integration here you are using $\displaystyle \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} $ this is a very well known region, it's circular region. When you intersect two cylinders, you get a square region as in the link.

The also note that the height function of this intersection of cylinders are different on different regions, as one cylinder's height is taller than other on some region. On region $(a)$ cylinder $(2)$ is taller than cylinder $(1)$ therefore cylinder $(1)$ acts are bounds for floor and ceiling.

And I prefer the method $(2)$.