   Chapter 13.2, Problem 57E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Total income The income from an oil change service chain can be considered as flowing continuously at an annual rate given by f ( t ) =   10 , 000 e 0.02 t ( d o l l a r s / y e a r ) Find the total income for this chain over the first 2 years (from t =   0  to  t =   2 ).

To determine

To calculate: The total income for the oil change service chain over the first 2 years where the income is given by f(t)=10,000e0.02t dollars per year.

Explanation

Given Information:

The provided function is f(t)=10,000e0.02t.

Formula used:

To calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

The value of the integral eaxdx=eaxa where a is constant and x is variable.

Calculation:

Consider the function f(t)=10,000e0.02t.

Since, total income from t=0 to t=2 is represented by an integral with limits from t=0 to t=2.

Thus, total income is 02f(t)dt.

Since, f(t)=10,000e0.02t.

Thus, the integral is 0210,000e0.02tdt.

Recall that to calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

Thus, evaluate the integral.

Consider the expression (0.02t).

Differentiate with respect to t.

d(0.02t)dt=0.02d(0.02t)=0.02dt

Thus, d(0.02t)=0.02dt.

Simplify the integral by the use of d(0.02t)=0.02dt and eaxdx=eaxa,

0210,000e0

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