Suppose we have a wire 20 inches long spread on the x-axis between 0 and 20.
At any point x, the density is 15 + 2x grams per linear centimeter.
Density varies between 15 grams to the left end (x = 0) and 55 grams has the right end (x = 20).
1.) Use amounts of left and right with n=10 and n=200 to yield an approximation of the weight of the wire.
2.) Install the appropriate limit and determine the integral that will precisely determine the mass of the wire.

Question

Suppose we have a wire 20 inches long spread on the x-axis between 0 and 20.
At any point x, the density is 15 + 2x grams per linear centimeter.
Density varies between 15 grams to the left end (x = 0) and 55 grams has the right end (x = 20).
1.) Use amounts of left and right with n=10 and n=200 to yield an approximation of the weight of the wire.
2.) Install the appropriate limit and determine the integral that will precisely determine the mass of the wire.

anonymous · Answer

First off this question doesn't make sense.  The wire should be 20cm long, NOT 20inches long because the density function is given in g/cm and d(20)=55 grams.  But assuming the length is given in cm's...An element of weight of the wire is dw=d*dx, where d=linear density which is given as d(x)=15+2x.  I'm not going to bother with the numerical approximation but the integral is setup as follows:|dw:1353642716602:dw|$$W=\int\limits_{0}^{20}(15+2x)dx=[15(20)+(20)^2-0]=700$$The total mass of the wire is 700 grams.