(a) By reading values from the given graph of f , use five...

(a) By reading values from the given graph of $ f $, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 10 $. In each case sketch the rectangles that you use. (b) Find new estimates using ten rectangles in each case.

(a) By reading values from the given graph of $ f $, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 10 $. In each case sketch the rectangles that you use.
(b) Find new estimates using ten rectangles in each case.

Key Concepts

Riemann Sums

A Riemann sum is a method for approximating the definite integral, or the area under a curve, by dividing the region into small subintervals and summing up areas of shapes (typically rectangles) that approximate the area over each subinterval. It provides a bridge between discrete summation and continuous integration.

Lower and Upper Approximations

Lower and upper approximations are techniques used in Riemann sums where the smallest and largest values of the function on each subinterval are used to form rectangles that either underestimate (lower sum) or overestimate (upper sum) the area under the curve. This helps to establish bounds within which the true area (integral) lies.

Partitioning of Intervals

Partitioning intervals involves dividing the interval over which the area is to be approximated into a finite number of equally or unequally spaced subintervals. The number of rectangles (or subintervals) affects the precision of the approximation; increasing the number generally leads to a more accurate estimate of the area.

Effect of Increasing the Number of Rectangles

Increasing the number of rectangles in the approximation generally improves accuracy because each rectangle covers a smaller portion of the area under the curve, thereby reducing the error inherent in the upper and lower approximations. This approach demonstrates how finer partitions lead to better estimates of the actual integral.

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