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The left-hand sum is a technique used in numerical integration to approximate the area under a curve. It is particularly useful when dealing with a function over a specific interval, like the velocity function provided in the exercise. The key idea is to create rectangles that approximate the area beneath the curve from the left endpoint of each subinterval.When calculating the left-hand sum, each rectangle's height is determined by the value of the function at the left endpoint of the subinterval. Here's how you can break it down:

• Division of Interval: First, divide the interval \( [a, b] \) into equal parts. The width of each subinterval is called \( \Delta t \).
• Height of Rectangles: For each subinterval, calculate the height using the function value at its left endpoint.
• Summing Those Areas: Multiply the height by \( \Delta t \) for each rectangle, then add up all these products.

For example, with the function \( v = 5 - 2t \) and \( n = 5 \), you divide the interval into five equal parts of width \( 0.4 \). The left-hand sum uses the function values at \[ t_0, t_1, t_2, t_3, \text{ and } t_4 \] to calculate an approximate value of the distance traveled.

The right-hand sum is another technique in numerical integration similar to the left-hand sum but uses the right endpoints of each subinterval to determine the height of the rectangles. This method can sometimes give a higher or lower estimate than the left-hand sum, depending on the function's behavior over the interval.Understanding the right-hand sum involves these steps:

• Interval Splitting: Just like before, split the interval \( [a, b] \) into equal subintervals, with width \( \Delta t \).
• Height of Rectangles: For each subinterval, use the value of the function at the right endpoint to define the rectangle's height.
• Calculate Total Area: Multiply each height by \( \Delta t \), and sum all these areas to get the right-hand sum.

In the exercise provided, for \( n = 5 \), the right-hand sum applies the function values at \[ t_1, t_2, t_3, t_4, \text{ and } t_5 \]. As noted in the computations, this can help in approximating the total distance traveled by the object over the interval.

Rectangular approximation is a versatile technique used in numerical integration to estimate the area under a curve, representing real-world quantities like distance when given a velocity-time graph. This method involves approximating the area under a curve by summing up the areas of multiple rectangles.Let's explore how rectangular approximation works:

• Subdivision of the Area: Start by dividing the curve's interval into smaller subintervals of equal width, \( \Delta t \). Each subinterval corresponds to the width of a rectangle.
• Choosing Rectangle Heights: Depending on whether using a left-hand or right-hand sum, choose the height of each rectangle based on the function values at specific points on the subinterval.
• Summing Rectangle Areas: Multiply the height and width of each rectangle to find their area, then sum all these areas to approximate the integral.

Rectangular approximation provides a simple yet powerful way to tackle real-life integration problems, like estimating distances when only discrete measurements are available. By increasing the number of rectangles (e.g., moving from \( n = 5 \) to \( n = 10 \)), the approximation becomes more accurate, as the rectangles provide a closer fit to the actual shape of the curve.