Areas and Volumes by Riemann Sums

From slicing a region to summing columns

What the integral computes

A direct computation on a small rectangle

Why the definition matters

The double integral $\iint_R f\, dA$ is defined as which limit?

Over a region of area $4$, the graph of $f$ sits above the plane on part of it with volume $9$ and below on the rest with mirrored depth $7$. What is $\iint_R f\, dA$?

What is $\iint_R 3\, dA$ over the rectangle $[0, 2] \times [0, 1]$?

Give $\iint_R 2\, dA$ where $R$ is the disk of radius $3$ centered at the origin, as a multiple of pi like 12*pi.

Using the midpoint rule with one cell, estimate $\iint_{[0,1] \times [0,1]} (x + y)\, dA$, giving the estimate as a plain number.