What is the formal mathematical definition of the double integral for volume derived from the limit of Riemann sums?

Conditions

• The function $f(x,y)$ represents the height of the surface above the region $D$.
• The limit is taken as the norm of the partition $||T||$ (maximum diameter of sub-regions) goes to zero.
• The sum extends over all $n$ sub-regions of the partition.

Reasoning, step by step

1. Identify the components of the Riemann sum: height $f(\xi_i, \zeta_i)$ and base area $\Delta\sigma_i$.
2. Formulate the summation $\sum_{i=1}^n f(\xi_i, \zeta_i)\Delta\sigma_i$.
3. Apply the limiting process where the partition becomes infinitely fine ($||T||\to0$).
4. Equate this limit to the double integral notation $\iint_D f(x,y)d\sigma$.
5. State the final result as the exact volume $V$.

Example

The video displays the formula: $V = \iint_D f(x,y)d\sigma = \lim_{||T||\to0} \sum_{i=1}^n f(\xi_i, \zeta_i)\Delta\sigma_i$. The script notes: 'Add their products and let the maximum mesh size tend to zero to obtain the exact volume.'

Common misconceptions

• Assuming the double integral is simply the sum of heights without weighting by area elements.
• Thinking that the limit applies to the number of subdivisions $n$ going to infinity independently of the mesh size shrinking to zero (though related, the rigorous definition requires the mesh norm to vanish).

Watch the explanation

Explore next

• Why is the condition of continuous nonnegative height justified for this specific example?
• How does this definition change if the surface dips below the xy-plane?