An oil tanker is leaking oil at the rate given (in barrels...

An oil tanker is leaking oil at the rate given (in barrels per hour) by $$L^{\prime}(t)=\frac{80 \ln (t+1)}{t+1}$$ where t is the time (in hours) after the tanker hits a hidden rock (when t = 0 ). a. Find the total number of barrels that the ship will leak on the first day. b. Find the total number of barrels that the ship will leak on the second day. c. What is happening over the long run to the amount of oil leaked per day?

An oil tanker is leaking oil at the rate given (in barrels per hour) by
$$L^{\prime}(t)=\frac{80 \ln (t+1)}{t+1}$$
where t is the time (in hours) after the tanker hits a hidden rock
(when t = 0 ).
a. Find the total number of barrels that the ship will leak on the first day.
b. Find the total number of barrels that the ship will leak on the second day.
c. What is happening over the long run to the amount of oil leaked per day?

Key Concepts

Substitution Method in Integration

The substitution method is an integration technique used to simplify integrals by changing variables. When a function involves compositions of functions or expressions like a logarithm combined with algebraic terms, substitution can reduce the integral to a more manageable form. This method is fundamental in solving integrals that do not conform to standard forms.

Asymptotic Analysis and Improper Integrals

Asymptotic analysis examines the behavior of functions as the independent variable approaches infinity or some critical value. In the context of accumulating quantities over an unbounded time interval, one might use the concept of improper integrals to determine if the accumulated quantity converges to a finite limit or diverges. This is vital in predicting long-run trends and behaviors in dynamic systems.

Rate of Change and Accumulation

This concept involves understanding that a rate of change, often given as a derivative, represents how a quantity is changing over time. By integrating the rate function, one can find the total accumulation of the quantity over a specified time period. This is a cornerstone of calculus that connects instantaneous rates (derivatives) with total change (integrals).

Definite Integration

Definite integration is the process of finding the integral of a function over a specific interval. In physical applications, it allows for the computation of the total accumulation (such as volume, mass, or, in similar problems, oil spilled) between two points in time. This is essential for quantifying exactly how much a quantity has changed over a finite duration.

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