The graph of f(t) is in Figure 5.26 . Which of the following...

The graph of $f(t)$ is in Figure $5.26 .$ Which of the following four numbers could be an estimate of $\int_{0}^{1} f(t) d t$ accurate to two decimal places? Explain your choice.
I. -98.35
II. 71.84
III. 100.12
IV. 93.47
(GRAPHS CANNOT COPY)

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Key Concepts

Definite Integral

A definite integral represents the net accumulation of a quantity over an interval and is interpreted as the net area under the curve of a function between two points. This concept is essential in connecting a function’s behavior with the total magnitude of its accumulated effect, taking into account areas above and below the axis.

Area Under the Curve

The area under a curve is a visual representation of the definite integral, where regions above the horizontal axis contribute positively and regions below contribute negatively. This concept is crucial in estimating the value of an integral by observing the overall balance of areas in a graph.

Estimating Integrals from Graphs

When an integral must be estimated from a graph, the process involves approximating the area under the function over a specified interval. This often involves visually assessing the heights and widths of regions under the curve and inferring a likely range for the accumulated area.

Accuracy and Rounding

Accuracy and rounding play key roles when numerical estimates are provided to a specific number of decimal places. Understanding how raw estimates are rounded and the potential effect on the final value is important in determining whether a given numerical value is a plausible estimate of the true integral.

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