How do you approximate the volume under $f(x,y)=9-x^2-y^2$ using four equal regions?

Conditions

• The base region is $[-2,2] \times [-2,2]$.
• The partition consists of four equal subsquares.
• Sample points are chosen according to a specific rule (e.g., bottom-left corners).

Reasoning, step by step

1. Partition $[-2,2] \times [-2,2]$ into four equal squares $A_1, A_2, A_3, A_4$.
2. Determine the dimensions of each subsquare: $\Delta x_k = 2$ and $\Delta y_k = 2$, so $\Delta A_k = 4$.
3. Choose sample points $(x_k,y_k)$ for each subsquare (e.g., $(-2,-2), (0,-2), (-2,0), (0,0)$).
4. Evaluate the function $f(x,y)=9-x^2-y^2$ at each sample point.
5. Sum the four function values.
6. Multiply the sum by the common area factor $2^2=4$ to get the approximate volume.

Example

The video works this example: using bottom-left corners $(-2,-2), (0,-2), (-2,0), (0,0)$, the function values are $1, 5, 5, 9$. The sum is $1+5+5+9=20$. Multiplying by the area $2^2=4$ gives an approximate volume of $80$. The slide shows: Volume $\approx [1+5+5+9]2^2 = 80$.

Common misconceptions

• Thinking the result 80 is the exact volume; the video emphasizes it is an approximation ($\approx$) because it uses only four boxes.
• Forgetting to multiply the sum of heights by the area of the subrectangles.
• Choosing sample points outside the subsquares.

Watch the explanation

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