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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!
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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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