The total number of acres of genetically modified crops...

The total number of acres of genetically modified crops grown worldwide from 1997 through 2003 was changing at the rate of $$ R(t)=2.718 t^{2}-19.86 t+50.18 \quad(0 \leq t \leq 6) $$ million acres/year. The total number of acres of such crops grown in 1997 ( $t=0$ ) was $27.2$ million acres. How many acres of genetically modified crops were grown worldwide in $2003 ?$

The total number of acres of genetically modified crops grown worldwide from 1997 through 2003 was changing at the rate of
$$
R(t)=2.718 t^{2}-19.86 t+50.18 \quad(0 \leq t \leq 6)
$$
million acres/year. The total number of acres of such crops grown in 1997 ( $t=0$ ) was $27.2$ million acres. How many acres of genetically modified crops were grown worldwide in $2003 ?$

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Key Concepts

Initial Value Problems

Initial value problems combine the use of a known starting value with the integration of a rate function. They are common in applications where a quantity changes over time, and the goal is to determine the final value after accounting for that change from the initial condition.

Polynomial Integration

Polynomial integration is a technique used to calculate antiderivatives of polynomial functions. This method is essential for solving problems involving rate functions that are expressed as polynomials, facilitating the computation of the accumulated total over a given interval.

Rate of Change

The rate of change represents the speed at which a quantity changes with respect to time. In mathematical contexts, it is the derivative of the quantity and provides an instantaneous measurement of change, which must be integrated over time to obtain the total accumulation.

Definite Integration and Accumulation

Definite integration involves calculating the area under a curve over a specified interval. In accumulation problems, such as finding a total quantity from a rate, the definite integral sums the small changes over time and, when combined with an initial condition, gives the overall quantity at a later time.

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