WORKSHEET X: Estimating double Integrals using
contour diagrams
I
(a) Using the following contour diagram, estimate the volume beneath the surface z = 2y and above
the rectangle R = [0, 3] [0, 3] in the xy-plane.
(b) Using a double integral, compute the exact value of this volume.
II (a) Using the following contour diagram, estimate the volume beneath the surface z = x + 2y and
above the rectangle R = [0, 4][0, 3] in the xy-plane.
(b) Using a double integral, compute the exact value of this volume.
III (a) Using the following contour diagram, estimate the volume beneath the surface
z = x sin(3x2) cos y and above the rectangle R = [0.4, 1] [-1.5, 1.5] in the xy-plane.
(b) Using a double integral, compute the exact value of this volume.
IV (a) Using the following contour diagram, estimate the volume beneath the surface
z = 7 – x2 – y3 + x and above the rectangle R = [-2, 3] [-2, 2] in the xy-plane.
FYI: Here is the Mathematica code that generates this contour diagram:
(b) Using a double integral, compute the exact value of this volume.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I
think it defines more unequivocally than anything else the inception of modern mathematics; and the system of
mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact
thinking.
- John von Neuman
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