Consider a continuous function \(f(x,y)≥0\) of two variables defined on the closed rectangle \(R\):
Let \(S\) be the solid that lies above \(R\) and under the graph of \(f\):
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We divide the region \(R\) into small rectangles \(R_{ij}\), each with area \(\Delta A\) and with sides \(\Delta x\) and \(\Delta y\). We do this by dividing the interval \([a,b]\) into \(m\) subintervals and dividing the interval \([c,d]\) into \(n\) subintervals. Hence \(\Delta x= \frac{b-a}{m}\), \(\Delta y= \frac{d-c}{n}\), and \(\Delta A=\Delta x \Delta y\). Let \((x^*_{ij},y^*_{ij})\) be an arbitrary sample point in each \(R_{ij}\).
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The volume of a thin rectangular box above \(R_{ij}\) is approximately \(f(x^*{ij},y^*_{ij}) \Delta A\):
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An approximate volume of the solid \(S\) is \(V ≈ \sum_{i=1}^m \sum_{j=1}^n f(x^*_{ij},y^*_{ij}) \Delta A\). This sum is known as a double Riemann sum. We obtain a better approximation to the actual volume if \(m\) and \(n\) become larger.
We use this to define the double integral: