In the applications that follow, it is helpful to sketch...

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary. The instantaneous rate of change in demand for U.S. lumber since $1970(t=0)$, in billions of cubic feet per year, is given by $$ Q^{\prime}(t)=12+0.006 t^{2} \quad 0 \leq t \leq 50 $$ Find the area between the graph of $Q^{\prime}$ and the $t$ axis over the interval $[35,40],$ and interpret the results.

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to interpret results. A graphing calculator will prove useful if you have one, but it is not necessary.
The instantaneous rate of change in demand for U.S. lumber since $1970(t=0)$, in billions of cubic feet per year, is given by
$$
Q^{\prime}(t)=12+0.006 t^{2} \quad 0 \leq t \leq 50
$$
Find the area between the graph of $Q^{\prime}$ and the $t$ axis over the interval $[35,40],$ and interpret the results.

Key Concepts

Definite Integrals

A definite integral represents the accumulated sum of infinitesimally small quantities over a specific interval. In calculus, it is used to calculate the total amount of a quantity, such as area or accumulated change, starting from a rate function by summing the contributions across an interval.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration by asserting that integration can be reversed by differentiation. It states that if you have a continuous function, then the definite integral over an interval can be found using any of its antiderivatives, which equates the integral to the net change in the original function over that interval.

Interpretation of Area Under a Curve

When a function represents an instantaneous rate of change, the area under its curve (or its definite integral) over a given interval corresponds to the total accumulated change of the underlying quantity during that period. This concept is foundational in interpreting how local changes aggregate to produce a global effect.

Instantaneous Rate of Change and Accumulation

An instantaneous rate of change describes how fast a quantity is changing at a specific moment in time. Integrating such a rate over a period provides a measure of how much the quantity has increased or decreased overall, offering a direct connection between the dynamics of change and the cumulative result.

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