Calculus I - Applications of the integral

2 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Average value of a function over an interval

   Suppose fff is a function that is continuous on the closed interval [a,b][a,b][a,b] . The average value yˉ\bar{y}yˉ​ of fff over [a,b][a,b][a,b] is yˉ=1b−a∫abf(x)dx=\bar{y}=\frac{1}{b-a} \int_a^b f(x)dx=yˉ​=b−a1​∫ab​f(x)dx= the value f(u)f(u)f(u) in the Mean Value Theorem for Integrals

2. Rectilinear motion

   for an object in rectilinear motion moving with velocity v=v(t)v=v(t)v=v(t) The net displacement of an object from [a,b][a,b][a,b] is given by ∫abv(t)dt\int_a^b v(t)dt∫ab​v(t)dt The total distance traveled by the object from [a,b][a,b][a,b] is given by ∫ab∣v(t)∣dt\int_a^b |v(t)|dt∫ab​∣v(t)∣dt If v(t)≥0v(t) \geq0v(t)≥0 for a≤t≤ba \leq t \leq ba≤t≤b , then the net displacement and total distance traveled are equal