Let f(x)=x^2. (a) Sketch the function over the interval...

Let $f(x)=x^{2}$. (a) Sketch the function over the interval [0,2] and the rectangles corresponding to $L_{4} .$ Calculate the area contained within them. (b) Sketch the function over the interval [0,2] again but with the rectangles corresponding to $R_{4} .$ Calculate the area contained within them. (c) Make a conclusion about the area under the curve $f(x)=x^{2}$ over the interval [0,2]

Let $f(x)=x^{2}$.
(a) Sketch the function over the interval [0,2] and the rectangles corresponding to $L_{4} .$ Calculate the area contained within them.
(b) Sketch the function over the interval [0,2] again but with the rectangles corresponding to $R_{4} .$ Calculate the area contained within them.
(c) Make a conclusion about the area under the curve $f(x)=x^{2}$ over the interval [0,2]

Key Concepts

Partition of an Interval

The process of dividing a given interval into smaller subintervals. This is an essential step in constructing Riemann sums, as the accuracy of the area approximation improves with finer partitions.

Right Endpoint Riemann Sum

Another form of Riemann sum where each rectangle’s height is determined by evaluating the function at the right endpoint of each subinterval. Like the left endpoint method, this technique provides an approximation of the area under a curve and will generally overestimate or underestimate the area depending on whether the function is increasing or decreasing.

Definite Integral

The limit of the Riemann sums as the number of subintervals approaches infinity, which yields the exact area under the curve over a specified interval. This concept is fundamental in calculus as it bridges the gap between discrete approximations and continuous areas.

Riemann Sum

A method for estimating the area under a curve by summing the areas of multiple rectangles whose heights are determined by the function's values at specific points within subintervals of the overall interval.

Left Endpoint Riemann Sum

A specific type of Riemann sum where the height of each rectangle is calculated using the function’s value at the left end of each subinterval. This approach is particularly useful for approximating areas when the function is increasing or decreasing, as it will systematically underestimate or overestimate the true area, respectively.