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Question: the method of quotslice approximate integrate can be used to...

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The method of Slice, Approximate, Integrate can be used to compute various geometric quantities - such as areas, volumes, and lengths - but it is also an important technique used in many physical applications as well. While the nature of the problems may be different; the method used to solve them is not! Suppose a spring has a spring constant3 k-|ON/m. Letz = 0 be the equilibrium position of the spring. Question 1: Set up and evaluate an integral that gives the total amount of work required to stretch the spring from 0 to z-4 The eract work required is: This formula for the work is actually obtained by the Slice, Approximate, Integrate procedure! To see this, we first recall some results from physics: Work for students comfortable with physics If you are comfortable with physics, you may think of work as follows. Under the assumptions: 1. The force F required to move a particle a distance d is constant. 2. The force F is in the direction of motion (which it always will be for us at this juncture of the course The work required to move a particle d units is given by: W = Fd. In the case of a spring, the force required to stretch the spring r meters from its equi- brium position is given by F(x) - kr, which is not constant The spring constant measures how dificult it is to stretch or compress the spring; the larger the constant, the more force is required to displace the spring from its equilibrium position!
Using this procedure, fill in the table below. You should include one sample calculation in the box provided after the table; it is not necessary to include all of the calculations! n (in m) Fu (in N) Wn in J) 2 30 30 Use this space to show how you obtained your values for n, Fn and Wn in the table for one value of n (other than n 3). The approximate amount of work required to stretch the spring frorn x = 0 to x the sum of works Wi, W2, Ws, and W4 you found above. is Question 1: What is the approximate amount of work required to stretch the spring? The approzimate work (in J) is: Is the answer close to the actual volume of the solid you computed at the beginning of the problem? How could we obtain a better approximation? The answer, as usual, is to use more slices! Of course, it would be a pain to do this by hand! Indeed, if we use 100 slices, we
would lwe to fiad the volus of 100dink in a similar to the alne and al Question 2: By uing the indicated uber ofcaculate the od work repaired to stwetch the spring from to the Projests Solder that gave inractions for muting the w ther e dd this comectly,male sure that yOEr W o4 andn-100 mach the ziven hese. W, (in J) 100 10 50 80.8 500 1000 gnt closet to this as n In this appeximation step, is it really a good te that Fis riation in Fi where Ar is the length of the intervl Thswn we take malt Note that we could find a formula that gives the agpewk in of over the the &-th isteral Tod esact result immedistely Step 3: Integrate We have detenmined that From this, we may imediately write dow ofthe rightmost infannímalduellere Fle)-kz, ๑-0,and
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