A what is the area of a thin slice of s at position x

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(a) What is the area of a thin slice of S at position x with width d x ? (b) What is the mass of a small piece of R at position x with length d x ? (c) What is the total area of S ? (d) What is the total mass of R ? (e) What is the x -coordinate of the centroid of S ? (f) What is the centre of mass of R ? In Questions 8 through 10 , you will derive the formulas for the centre of mass of a rod of variable density, and the centroid of a two-dimensional region using vertical slices (Equations 2.3.2 and 2.3.3 in the CLP–II text). Knowing the equations by heart will allow you to answer many questions in this section; understanding where they came from will you allow to generalize their ideas to answer even more questions. 75
A PPLICATIONS OF I NTEGRATION 2.3 C ENTRE OF M ASS AND T ORQUE Q[8]: Suppose R is a straight, thin rod with density ρ ( x ) at a position x . Let the left endpoint of R lie at x = a , and the right endpoint lie at x = b . (a) To approximate the centre of mass of R , imagine chopping it into n pieces of equal length, and approximating the mass of each piece using the density at its midpoint. Give your approximation for the centre of mass in sigma notation. m 1 m 2 m n a b (b) Take the limit as n goes to infinity of your approximation in part (a), and express the result using a definite integral. Q[9]: Suppose S is a two-dimensional object and at (horizontal) position x its height is T ( x ) ´ B ( x ) . Its leftmost point is at position x = a , and its rightmost point is at position x = b . To approximate the x -coordinate of the centroid of S , we imagine it as a straight, thin rod R , where the mass of R from a ď x ď b is equal to the area of S from a ď x ď b . (a) If S is the sheet shown below, sketch R as a rod with the same horizontal length, shaded darker when R is denser, and lighter when R is less dense. x y T ( x ) B ( x ) a b (b) If we cut S into strips of very small width d x , what is the area of the strip at position x ? (c) Using your answer from (b), what is the density ρ ( x ) of R at position x ? (d) Using your result from Question 8 ( b ), give the x -coordinate of the centroid of S . Your answer will be in terms of a , b , T ( X ) , and B ( x ) . Q[10]: Suppose S is flat sheet with uniform density, and at (horizontal) position x its height is T ( x ) ´ B ( x ) . Its leftmost point is at position x = a , and its rightmost point is at position x = b . To approximate the y -coordinate of the centroid of S , we imagine it as a straight, thin, vertical rod R . We slice S into thin, vertical strips, and model these as weights on R with: position y on R , where y is the centre of mass of the strip, and mass in R equal to the area of the strip in S . (a) If S is the sheet shown below, slice it into a number of vertical pieces of equal length, 76
A PPLICATIONS OF I NTEGRATION 2.3 C ENTRE OF M ASS AND T ORQUE approximated by rectangles. For each rectangle, mark its centre of mass. Sketch R as a rod with the same vertical height, with weights corresponding to the slices you made of S .

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