15.1+note+outline.pdf

15.1+note+outline.pdf

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1 15.1 Double Integrals over Rectangles We want to generalize the idea behind integrals in order to compute volumes. For single variable functions f : ! ! , what is the interpretation of a definite integral? Subdivide the interval [ , ] a b and form a Reimann Sum n R : When we consider lim n n R →∞ we get: Volume and Double Integrals Let ( , ) f x y be a two variable function. Consider a rectangular region R in the domain of f. R = S is the solid bounded by the rectangle R and the graph of f in the region R . To approximate the volume of S , subdivide R into _____________________________. Choose a sample point to determine the height of the ___________________________.
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2 An approximation of the volume in each region is given by: And an approximation of the volume of S is: This is called a ______________________________________________________. The smaller the rectangles in the domain, the better the approximation is of the volume. The actual volume is… We can compute integrals by first setting up a Reimann sum and then finding the limit. For now, we only want to approximate the limit using the midpoint of each sub-rectangle.
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