EXAMPLE 1 Estimate the volume of the solid that lies above the square R = [o, 10

Question

EXAMPLE 1 Estimate the volume of the solid that lies above the square R = [o, 10] x [o, 10] and below the elliptical paraboloid z502 - x2 - 4y2. Divide R into 4 equal squares and choose the sample point to be the upper right corner of each square Rif. Sketch the solid and the approximating rectangular boxes. The paraboloid is the graph of f(x, y) 502 - x* - 4y2 and the area of each SOLUTION square is 25 . Approximating the volume of the Riemann sum with m n 2, we have 3772577(25) x (25) +2(25)- This is the volume of the approximating rectangular boxes shown in the figure.

Explanation / Answer

f(5,5) = 502-52-4(5)2=377

f(5,10)= 502-52-4(102)=77

f(10,5) = 502 - 102-4(52) =302

f(10,10)= 502-102-4(10)2=2

So we will get

377(25)+77(25)+302*(25)+2(25)= 18950

This is the volume of the approximting rectangular boxes shown in the figure .

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