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NAME
APPROXIMATING DOUBLE INTEGRALS
Recall that in Calculus I and Calculus II, there are some definite integrals for
which we cannot obtain the exact value. The best we can do is approximate the
value of the definite integral. We can have the same situation with double (or
triple) integrals. This procedure was discussed in Section 16.1. Suppose we wish
to approximate the integral
\int_{x=-1}^{x=1} \int_{y=0}^{y=1} \sin(\sqrt{x+1+y}) \, dy \, dx
(e) For each subrectangle, find the coordinates \((x, y)\) of the lower right
corner and list them.
(f) We will approximate the double integral using sums, similar to what we
have done in the past.
\int_{x=-1}^{x=1} \int_{y=0}^{y=1} \sin(\sqrt{x+1+y}) \, dy \, dx \approx \sum_{k=1}^{mn} \sin(\sqrt{x_k+1+y_k}) \Delta x \Delta y
Using the points from part (e), approximate the double integral.
BE SURE YOUR CALCULATOR IS IN RADIAN MODE.
NOTE: DO NOT round until the end, then round your answer to five
decimal places. Show ALL your work for the calculation below.
