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The double
∬Rf(x,y)dA=limn→+∞n∑k=1f(x*k,y*k)∇Ak
Describe the procedure on which this definition is based.
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The procedure on which the double integral over a region R in the xy−plane is defined as ∬Rf(x,y)dA=limn→+∞n∑k=1f(x∗k,y∗k)ΔAk is based.
Answer to Problem 1RE
If a region divided into n regions with lesser size and with approximately equal areas and the number of partitions increases, the approximation is usually better.
Explanation of Solution
Consider the given double integral over a region R in the xy−plane is defined as ∬Rf(x,y)dA=limn→+∞n∑k=1f(x∗k,y∗k)ΔAk .
First, compute a double integral ∬Rf(x,y)dA over region R .
Next, divide the region R into n regions with lesser size and with approximately equal areas ΔAk .
Now, approximate the integral adding the products of the areas multiplied by the values of the given function f(x,y) evaluated at some point within this region (x∗k,y∗k) .
∬Rf(x,y)dA≈limn→+∞n∑k=1f(x∗k,y∗k)ΔAk
When the number of partitions increases, the approximation is usually better.
Therefore, if the sum converges, when the number of partitions tends to infinity, then the exact result is obtained.
∬Rf(x,y)dA=limn→+∞n∑k=1f(x∗k,y∗k)ΔAk
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