# D the finite region s in the first quadrant bounded

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Chapter 7 / Exercise 21
Calculus
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(d) The finite regionSin the first quadrant bounded by the curvesy= 2xandy=x3;revolveSabout they-axis.(e) The finite regionSbounded by the curvesx= (y-1)2andy=x-1; revolveSaboutthey-axisC
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Chapter 7 / Exercise 21
Calculus
Stewart Expert Verified
1426.2. USING DEFINITE INTEGRALS TO FIND VOLUMEActivity 6.6.In each of the following questions, draw a careful, labeled sketch of the region described, aswell as the resulting solid that results from revolving the region about the stated axis. In addi-tion, draw a representative slice and state the volume of that slice, along with a definite integralwhose value is the volume of the entire solid. It is not necessary to evaluate the integrals youfind.For each prompt, use the finite regionSin the first quadrant bounded by the curvesy= 2xandy=x3.(a) RevolveSabout the liney=-2.(b) RevolveSabout the liney= 4.(c) RevolveSabout the linex=-1.(d) RevolveSabout the linex= 5.C
6.3. DENSITY, MASS, AND CENTER OF MASS1436.3Density, Mass, and Center of MassPreview Activity 6.3.In each of the following scenarios, we consider the distribution of a quantityalong an axis.(a) Suppose that the functionc(x) = 200 + 100e-0.1xmodels the density of traffic on a straightroad, measured in cars per mile, wherexis number of miles east of a major interchange,and consider the definite integralR20(200 + 100e-0.1x)dx.i. What are the units on the productc(x)· 4x?ii. What are the units on the definite integral and its Riemann sum approximation givenbyZ20c(x)dxnXi=1c(xi)4x?iii. Evaluate the definite integralR20c(x)dx=R20(200 + 100e-0.1x)dxand write one sen-tence to explain the meaning of the value you find.(b) On a 6 foot long shelf filled with books, the functionBmodels the distribution of theweight of the books, measured in pounds per inch, wherexis the number of inches fromthe left end of the bookshelf. LetB(x)be given by the ruleB(x) = 0.5 +1(x+1)2.i. What are the units on the productB(x)· 4x?ii. What are the units on the definite integral and its Riemann sum approximation givenbyZ3612B(x)dxnXi=1B(xi)4x?iii. Evaluate the definite integralR720B(x)dx=R720(0.5 +1(x+1)2)dxand write one sen-tence to explain the meaning of the value you find../
1446.3. DENSITY, MASS, AND CENTER OF MASSActivity 6.7.Consider the following situations in which mass is distributed in a non-constant manner.(a) Suppose that a thin rod with constant cross-sectional area of 1 cm2has its mass dis-tributed according to the density functionρ(x) = 2e-0.2x, wherexis the distance in cmfrom the left end of the rod, and the units onρ(x)are g/cm. If the rod is 10 cm long,determine the exact mass of the rod.(b) Consider the cone that has a base of radius 4 m and a height of 5 m. Picture the conelying horizontally with the center of its base at the origin and think of the cone as a solidof revolution.
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