   Chapter 5.5, Problem 85E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Dialysis treatment removes urea and other waste products from a patient’s blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often well described by the equation u ( t ) = r V C 0 e − r t / V where r is the rate of flow of blood through the dialyzer (in mL/min), V is the volume of the patient’s blood (in mL), and C0 is the amount of urea in the blood (in mg) at time t = 0. Evaluate the integral ∫ 0 30 u ( t )   d t and interpret it.

To determine

To evaluate: The integral and interpret the value.

Explanation

Given:

Describe the Equation which the rate of removal of urea from the blood as follows;

u(t)=rVC0ertV.

Calculation:

Show the integral function.

030u(t)dt

Substitute (rVC0ertV) for u(t).

030u(t)dt=030rVC0ertVdt (1)

The region lies between t=0 and t=30.

Take the consideration as follows:

u=rtV (2)

Differentiate both sides of the Equation (2).

du=rVdtdu=rVdt

Calculate the lower limit value of u using Equation (2).

Substitute 0 for t in Equation (2).

u=r(0)V=0

Calculate the upper limit value of u using Equation (2).

Substitute 30 for t in Equation (2).

u=r(30)V=30rV

Apply lower and upper limits for u in Equation (1).

Substitute u for (rtV) and (du) for (rVdt) in Equation (1)

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