integration problem?

[匿名]

2016-03-28 12:52 pm

Suppose that the velocity of a vehicle at time t is given by the function
f(t) = 20t + 10t^2, 0 ≤ t ≤ 2
80 − 5(2 − t)^2, 2 < t ≤ 6
Find the distance travelled from t = 0 to t = 6.

回答 (1)

Lopez

2016-03-28 1:52 pm

✔ 最佳答案

distance
= ∫ v dt , from t = 0 to t = 6 , where v denotes velocity
= ∫ f(t) dt , from t = 0 to t = 6

∫ f(t) dt , from t = 0 to t = 2
= ∫ ( 20t + 10t^2 ) dt
= [ 10t^2 + (10/3)t^3 ] , from t = 0 to t = 2
= 10*2^2 + (10/3)*2^3
= 40 + 80/3
= 200/3

∫ f(t) dt , from t = 2 to t = 6
= ∫ [ 80 − 5(2 − t)^2 ] dt
= ∫ ( 80 − 20 + 20t - 5t^2 ) dt
= ∫ ( 60 + 20t - 5t^2 ) dt
= [ 60t + 10t^2 - (5/3)t^3 ] , from t = 2 to t = 6
= [ 60*6 + 10*6^2 - (5/3)6^3 ] - [ 60*2 + 10*2^2 - (5/3)2^3 ]
= 360 + 360 - 360 - 120 - 40 + 40/3
= 200 + 40/3
= 640/3

distance
= 200/3 + 640/3
= 840/3
= 280 ..... Ans

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