Some computer algebra systems have commands that will draw...

Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $x_{1}^{*}$ is a left or right endpoint. (For instance. in Maple use leftbox, $x$ ightbox, lefts um, and rightsum.) (a) If $f(x)=1 /\left(x^{2}+1\right), 0 \leqslant x \leqslant 1,$ 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 0.780 and 0.791.

Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $x_{1}^{*}$ is a left or right endpoint. (For instance. in Maple use leftbox, $x$ ightbox, lefts um, and rightsum.)
(a) If $f(x)=1 /\left(x^{2}+1\right), 0 \leqslant x \leqslant 1,$ 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 0.780 and 0.791.

Key Concepts

Riemann Sums

Riemann sums are a fundamental concept in calculus for approximating the area under a curve by dividing the region into simple shapes, usually rectangles, and summing their areas. The two common variants discussed are left and right Riemann sums, where the function is evaluated at either the left or right endpoints of each subinterval. This concept is essential for understanding how numerical integration approximates definite integrals.

Numerical Integration

Numerical integration involves approximating the value of a definite integral when an exact analytic solution is difficult or impossible to obtain. Techniques such as Riemann sums, the trapezoidal rule, and Simpson's rule fall under this category. These methods are particularly useful in computational applications and in cases where the function is given in tabulated form or defined by data points.

Approximation Error and Bounds

When using methods like left and right Riemann sums to approximate integrals, the true value of the integral is often bounded between these approximations. This yields valuable error estimates, informing us about the maximum possible deviation of the numerical approximation from the actual integral value. Understanding these error bounds is crucial in numerical analysis and computational mathematics.

Graphical Visualization of Integration

Graphical visualization helps in understanding numerical integration methods by illustrating how the subintervals of the domain are partitioned and how the corresponding rectangle areas contribute to the approximation of the total area under the curve. Such visual tools assist in grasping the geometric interpretation of integrals and the impact of choosing left or right endpoints on the approximation.

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