[11.2] - The shape of the solid
Each double integral describes the volume of a geometrical shape. Describe (and/or sketch) the shape. Use geometry to calculate the volume corresponding to the double integral.
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$\int\int_{[0,1]\times[0,1]}(1-y)\,dy\,dx$
The plane $1-y$ slices diagonally, halfway through a $1\times 1\times 1$ cube. So the volume is $\color{blue}1/2$.
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$\int_{-3}^3 \int_{y=-2}^2 \sqrt{4-y^2}\,dy\,dx$
$z=\sqrt{4-y^2}$ describes a half circle of radius=2 in the $yz$ plane. Overall this is one half of a cylinder of radius 2 stretching 6 units from -3 to 3 along the $x$ axis. The volume is $$V=\frac12 6*\pi*2^2=12\pi\approx \color{blue}37.7$$
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$\int_{-1}^1\int_{y=-2}^3( 1-|x|) \,dy\,dx$
A triangular prism: the cross section is a triangle of area 1 stretching 5 units from $y=-2$ to $3$. The volume is $1\cdot 5= \color{blue}5$.