In Exercises 19-22, calculate the Riemann sum R(f, P, C) for...

In Exercises $19-22,$ calculate the Riemann sum $R(f, P, C)$ for the given function, partition, and choice of sample points. Also, sketch the graph of $f$ and the rectangles corresponding to $R(f, P, C)$.
$$
f(x)=x^{2}+x, \quad P=\{2,3,4.5,5\}, \quad C=\{2,3.5,5\}
$$

Key Concepts

Riemann Sum

A Riemann sum is an approximation of the area under a curve (or the definite integral) found by dividing the interval of integration into subintervals, computing the function’s value at selected points in these subintervals, and then summing the areas of the corresponding rectangles. This method allows one to estimate the integral by using finite sums, and it forms the basis for the formal definition of the Riemann integral.

Partition and Sample Points

A partition divides the interval of interest into subintervals by selecting a set of points, while sample points (or evaluation points) are chosen within each subinterval to determine the height of each rectangle in the Riemann sum. Together, these concepts define how the overall summing process is structured and highlight the importance of interval selection and function evaluation in approximating the area under a curve.

Graphical Interpretation of Riemann Sums

Graphically representing a Riemann sum involves sketching the function along with the rectangles corresponding to each subinterval. Each rectangle’s width is the length of the subinterval, and its height is the value of the function at the chosen sample point. This visual approach helps in understanding how the Riemann sum approximates the area under the function’s curve and underscores the connection between the discrete sum and the continuous area being estimated.

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