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Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral.

12 π R 2 = 8 π R 2 R r d r

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Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral.

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Suppose that a particle moves along a straight line with velocity v ( t ) = 4 2 t , where 0 t 2 (in meters per second). Find the displacement at time t and the total distance traveled up to t = 2 .

d ( t ) = 0 t v ( s ) d s = 4 t t 2 . The total distance is d ( 2 ) = 4 m .

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Suppose that a particle moves along a straight line with velocity defined by v ( t ) = t 2 3 t 18 , where 0 t 6 (in meters per second). Find the displacement at time t and the total distance traveled up to t = 6 .

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Suppose that a particle moves along a straight line with velocity defined by v ( t ) = | 2 t 6 | , where 0 t 6 (in meters per second). Find the displacement at time t and the total distance traveled up to t = 6 .

d ( t ) = 0 t v ( s ) d s . For t < 3 , d ( t ) = 0 t ( 6 2 t ) d t = 6 t t 2 . For t > 3 , d ( t ) = d ( 3 ) + 3 t ( 2 t 6 ) d t = 9 + ( t 2 6 t ) . The total distance is d ( 6 ) = 9 m .

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Suppose that a particle moves along a straight line with acceleration defined by a ( t ) = t 3 , where 0 t 6 (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to t = 6 if v ( 0 ) = 3 and d ( 0 ) = 0 .

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A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/sec 2 . Neglecting air resistance, solve for the velocity v ( t ) and the height h ( t ) of the ball t seconds after it is thrown and before it returns to the ground.

v ( t ) = 40 9.8 t ; h ( t ) = 1.5 + 40 t 4.9 t 2 m/s

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A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is −9.8 m/sec 2 . Neglecting air resistance, solve for the velocity v ( t ) and the height h ( t ) of the ball t seconds after it is thrown and before it returns to the ground.

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The area A ( t ) of a circular shape is growing at a constant rate. If the area increases from 4 π units to 9 π units between times t = 2 and t = 3 , find the net change in the radius during that time.

The net increase is 1 unit.

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A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from 36 π in. 3 to 288 π in. 3 between time t = 30 and t = 60 seconds, find the net change in the radius of the balloon during that time.

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Water flows into a conical tank with cross-sectional area πx 2 at height x and volume π x 3 3 up to height x . If water flows into the tank at a rate of 1 m 3 /min, find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min.

At t = 5 , the height of water is x = ( 15 π ) 1 / 3 m . . The net change in height from t = 5 to t = 10 is ( 30 π ) 1 / 3 ( 15 π ) 1 / 3 m.

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A horizontal cylindrical tank has cross-sectional area A ( x ) = 4 ( 6 x x 2 ) m 2 at height x meters above the bottom when x 3 .

  1. The volume V between heights a and b is a b A ( x ) d x . Find the volume at heights between 2 m and 3 m.
  2. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, d x d t = d x d V d V d t , at how many meters per minute is the height of oil in the tank changing, expressed in terms of x , when the height is at x meters?
  3. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?
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Practice Key Terms 1

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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