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Evaluate the following integral:
(a) analytically, (b) single application of the trapezoidal rule, (c) composite trapezoidal rule with n = 2 and 4, (d) single application of Simpson’s 1/3 rule, (e) composite Simpson’s 1/3 rule with n = 4, (f) Simpson’s 3/8 rule, and (g) composite Simpson’s rule, with n = 5. For each of the numerical estimates (b) through (g), determine the true percent relative error based on (a).
In an experiment, a rectangular block with height h is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submerged. In liquids A, B, and C, it floats with heights h/2, 2h/3, and h/4 above the liquid surface, respectively. What is the relative densities (the densities relative to that of water) of B.