The velocity of a car was read from its speedometer at...

The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. $$\begin{array}{|c|c||c|c|}\hline t(s) & v(\mathrm{mi} / \mathrm{h}) & t(s) & v(\mathrm{mi} / \mathrm{h}) \\\hline 0 & 0 & 60 & 56 \\10 & 38 & 70 & 53 \\20 & 52 & 80 & 50 \\30 & 58 & 90 & 47 \\40 & 55 & 100 & 45 \\50 & 51 & & \\\hline\end{array}$$

The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car.
$$\begin{array}{|c|c||c|c|}\hline t(s) & v(\mathrm{mi} / \mathrm{h}) & t(s) & v(\mathrm{mi} / \mathrm{h}) \\\hline 0 & 0 & 60 & 56 \\10 & 38 & 70 & 53 \\20 & 52 & 80 & 50 \\30 & 58 & 90 & 47 \\40 & 55 & 100 & 45 \\50 & 51 & & \\\hline\end{array}$$

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical integration method that approximates the value of a definite integral by taking the value of the function at the midpoints of subintervals and multiplying by the subinterval width. This method provides a more accurate approximation when the function is relatively smooth over each interval.

Numerical Integration

Numerical integration encompasses techniques for approximating the value of an integral when an analytical solution is difficult or impossible to obtain. Methods such as the Midpoint Rule, trapezoidal rule, and Simpson's rule are commonly used to estimate areas under curves based on discrete data points.

Riemann Sum

A Riemann sum is a fundamental concept in calculus that approximates the total area under a curve by summing up areas of simple shapes like rectangles. The Midpoint Rule is a specific type of Riemann sum where the height of each rectangle is determined by the function value at the midpoint of each subinterval.

Unit Conversion

Unit conversion is the process of converting measurements from one set of units to another to maintain consistency in calculations, especially when combining quantities like speed and time. This is crucial in problems involving physical quantities, such as converting a velocity measurement into a compatible unit system for integration to obtain distance.

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