In this section we use definite integrals to study rectilinear motion and compute average value.
1 Applications of Definite Integrals
1.1 Average Value
The average value is given by \begin{align*} f_{ave} &= \frac{1}{b-a}\int _a^b f(x) \ dx \\ & = \frac 12\int _0^2 x^2 \ dx \\ & = \frac{x^3}{6} \Bigg |_0^2 \\ & = \tfrac 43 \end{align*}
The average value is given by \begin{align*} f_{ave} &= \frac{1}{b-a}\int _a^b f(x) \ dx \\ & = \frac{1}{\pi }\int _0^\pi \sin (x) \ dx \\ & = -\frac{1}{\pi } \cos (x) \Bigg |_0^\pi \\ & = \frac{2}{\pi } \end{align*}
The average value is given by \begin{align*} f_{ave} &= \frac{1}{b-a}\int _a^b f(x) \ dx \\ & = \frac 12\int _{-1}^1 e^{2x} \ dx \\ & = \frac 14 e^{2x} \Bigg |_{-1}^1 \\ & = \frac{e^2 - e^{-2}}{4} \end{align*}
1.2 Rectilinear Motion
Suppose that a projectile is launched vertically into the air. Then, the height of the object at time , the velocity at time , and the acceleration at time are related as follows: and
We can also use indefinite integrals to express these relationships: and
- (a)
- the displacement over the interval is given by and
- (b)
- the distance traveled over the interval is given by
- (a)
- Find a formula, , for the height of the object at time seconds.
- (b)
- Find the displacement of the object from time to time .
- (c)
- Find the total distance traveled from time to time .
Assuming that , as it is on the surface of the earth, we can compute by using an indefinite integral: To find the value of the constant , we use the fact that the initial velocity was given as : which implies that . Thus Next, we repeat the process to find : To find , we use the given information about the initial height of the object: which implies . Thus, Next, to find the displacement from to , we compute Note that we could have computed as a definite integral: since is an anti-derivative of . Finally, to find the total distance traveled from to time , we need integrate . To get a handle on , recall Noting that , we can see that
Hence, the total distance traveled from to is
- (a)
- Find a formula, , for the height of the object at time seconds.
- (b)
- Find the maximum height reached by the object.
- (c)
- Find the displacement of the object from time to time .
- (d)
- Find the total distance traveled from time to time .
Assume .
and
- (a)
- Find a formula, , for the height of the object at time seconds.
- (b)
- Find the maximum height reached by the object.
Assume .
and