The table shows values of a force function $ f(x) $, where $ x $ is measured in meters and $ f(x) $ in newtons. Use the Midpoint Rule to estimate the work done by the force in moving an object from $ x = 4 $ to $ x = 20 $.
W \approx 120.8
Applications of Integration
in this problem. The table shows values of force function. That is epics. Where X is measured in meters and epics in newtons. Okay, so we want to use made point rule to estimate the work done by the close in moving an object from X equals to 42 X equals to 20. So this is partial that the force function uh Right development is given by forced into displacement. So right, we are going to use that only. But let us say this interval is given to us. That is from four to 20 days to four x Forest between x equals to four and 20 shape. So let us say n equals to four means it has four sub intervals. Right? So both of these have intervals will be work. This is nothing. But this is all you can say since delic is given by b minus a upon end. Right? So this is midpoint rule uhh that in which sovereignty levels are divided. This is the birth of the internal some interviews you can see. So here the width of the of interdependent people, there will be four sub intelligence. So now seven devils will be but there is uh X equals to four. Then we have the difference between uh to some intervals you can see uh two values you can save values of X. It should be four. So 488, 12, 12, 16 and 16 20. Okay, so, mid points of these intervals are nothing. But the values between let's say this is let's say we have taken the sub interval 48. So the value will be made points will be what made points of these sub intervals will be bought. Excellent. Start. It is represented by we are representing it by start. Right, So that will be six and four seconds of interval. That is eight. It should be The midpoint is 10. Now for 12-16 the midpoint is 14 and for 16-20 the midpoint is nothing but this is it. Right. So now we can estimate the work done by the midpoint rollers force introduced placement. So forth, let us say this is F X I I start. Right? So, why we're writing this this is the midpoint. We are we're taking midpoint. Right? All right. So now we can see here. Okay, water things lettuce. Right. So this is the work done. Work done is given by nothing, but this is given by W equals to the legs plus the legs in true bracket. Right? In bracket. The sum of all the functions of eggs. Right? X. I start right, that is midpoint. So, we are writing it out down Further. This is FX one star plus fx to start. Fxc star plus I. F exports start. Right? So, for experts start that is that is six. Mid 26. So for 60 value is 5.84 fx Right, So, we will put all the values here in this way. So for 10 it is a point eight and for three, I'm sorry for 14 it should be 8.2, and for 18 it should be 5.2. So all that value we have written here. So finally we will get developed and us 112 to so this is how we solve this problem. I hope you understood the concept. Thank you for watching.