The vertical distribution of velocity of the water in a...

The vertical distribution of velocity of the water in a river may be approximated by $t=c(d-y)^{16},$ where $v$ is the velocity (in $\mathrm{m}$ sec) at a depth of $y$ meters below the water surface, $d$ is the depth of the river, and $c$ is a positive constant (a) Find a formula for the average velocity $v_{\text {av }}$ in terms of $d$ and $c$. (b) If $v_{0}$ is the velocity at the surface, show that $v_{\mathrm{av}}=\frac{6}{7} v_{0}$.

The vertical distribution of velocity of the water in a river may be approximated by $t=c(d-y)^{16},$ where $v$ is the velocity (in $\mathrm{m}$ sec) at a depth of $y$ meters below the water surface, $d$ is the depth of the river, and $c$ is a positive constant
(a) Find a formula for the average velocity $v_{\text {av }}$ in terms of $d$ and $c$.
(b) If $v_{0}$ is the velocity at the surface, show that $v_{\mathrm{av}}=\frac{6}{7} v_{0}$.

Key Concepts

Average Value of a Function

This concept refers to finding a single value that represents the typical value of a function over an interval. In applications like calculating the average velocity of water in a river, it involves integrating the velocity function over the depth of the river and then dividing by the total depth, thus providing a meaningful average for quantities that vary continuously.

Definite Integration

Definite integration is the process of finding the area under the curve of a function between two specific limits. It is a key tool in calculus that helps accumulate variable quantities over an interval, and it is used here to determine the overall contribution of varying velocity across the depth of the river.

Integration by Substitution

This technique simplifies an integral by changing the variable of integration, often making the integral easier to evaluate. It is particularly useful when the integrand contains composite functions or when a substitution can transform a complicated expression into a basic power function, as seen in problems involving power functions over a variable domain.

Power Rule for Integration

The power rule provides a method for integrating functions of the form x^n. It states that the integral of x^n with respect to x is x^(n+1)/(n+1), provided n is not equal to -1. This rule is essential for integrating polynomial and power functions, and it facilitates the calculation of integrals involving expressions raised to a power, such as those used in modeling variable velocity profiles.

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