(a) If f(x)=x /(x+2), 1 ⩽x ⩽4, use the commands discussed in...

(a) If $f(x)=x /(x+2), 1 \leqslant x \leqslant 4$, use the commands discussed in Exercise 11 to find the left and right sums for $n=10,30$, and 50 .
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $f$ lies between 1.603 and 1.624 .

Key Concepts

Riemann Sums

A method to approximate the area under a curve by summing the areas of a series of rectangles. As the number of rectangles increases, the approximation converges to the true area represented by the definite integral.

Left Endpoint Approximation

This technique computes the area of each rectangle based on the function value at the left endpoint of each subinterval. It is a type of Riemann sum, and its accuracy depends on the function's behavior in the interval.

Right Endpoint Approximation

Similar to the left endpoint method, but it uses the function value at the right endpoint of each subinterval to determine the rectangle heights. This method provides an alternate estimation for the area under the curve.

Partitioning the Interval

Dividing the integration interval into smaller subintervals (using a specified number of rectangles) is crucial in Riemann sums. The refinement of the partition (i.e., using more subintervals) generally leads to a more accurate approximation of the integral.

Definite Integral as the Limit of Riemann Sums

The exact area under a continuous curve between two points is defined as the limit of the Riemann sums as the number of subintervals approaches infinity. This concept bridges discrete approximations with continuous integration.

Graphical Representation of Riemann Sums

Illustrating the approximation through drawn rectangles over the curve provides a visual understanding of how the Riemann sums approximate the area under the curve, and how increasing the number of rectangles improves the estimate.

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