A rectangular prism with a length of 14 meters, width of 5 meters, and height of 4.5 meters. A square pyramid with a base of 8 f
eet by 8 feet and height of 6 feet.
Find the volumes of the three-dimensional figures.
Rectangular prism:
Pyramid:
Find the volumes of the three-dimensional figures.
Rectangular prism:
Pyramid:
2 answers:
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First, recall that Gaussian quadrature is based around integrating a function over the interval [-1,1], so transform the function argument accordingly to change the integral over [1,5] to an equivalent one over [-1,1].
So,
Let
. With 
, we're looking for coefficients 
and nodes 
, with 
, such that
You can either try solving for each with the help of a calculator, or look up the values of the weights and nodes (they're extensively tabulated, and I'll include a link to one such reference).
Using the quadrature, we then have
So,
Let
You can either try solving for each with the help of a calculator, or look up the values of the weights and nodes (they're extensively tabulated, and I'll include a link to one such reference).
Using the quadrature, we then have