Consider the area between the graph of f(x)=√(x) and the x -axis on [0,4]. (a) Use the four rect

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Consider the area between the graph of f(x)=√(x) and the x...

Consider the area between the graph of $f(x)=\sqrt{x}$ and the $x$ -axis on $[0,4]$. (a) Use the four rectangles shown on the left to approximate the given area, and then use the eight rectangles shown on the right to obtain another approximation of that area. Be sure to use the fact that the graph shown is that of the function $f(x)=\sqrt{x}$ in your calculations. (b) Describe what would happen if we did similar approximations with more and more rectangles, and make a guess for the resulting limit.

Consider the area between the graph of $f(x)=\sqrt{x}$ and the $x$ -axis on $[0,4]$.
(a) Use the four rectangles shown on the left to approximate the given area, and then use the eight rectangles shown on the right to obtain another approximation of that area. Be sure to use the fact that the graph shown is that of the function $f(x)=\sqrt{x}$ in your calculations.
(b) Describe what would happen if we did similar approximations with more and more rectangles, and make a guess for the resulting limit.

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by dividing the region into a finite number of rectangles. The sum of the areas of these rectangles, calculated using sample points in each subinterval, provides an approximation of the definite integral, and serves as the foundation for understanding the limit process in integration.

Definite Integral

The definite integral of a function over an interval represents the exact accumulated area under the curve between the bounds. It is formally defined as the limit of the Riemann sums as the number of rectangles approaches infinity, allowing for an exact calculation of area that approximations seek to estimate.

Partitioning of an Interval

Partitioning an interval refers to dividing the interval into a finite number of smaller subintervals. This is a crucial step in forming Riemann sums, as it determines the width of the rectangles; the accuracy of the approximation improves with finer partitions, meaning smaller subintervals.

Limit Process in Integration

The limit process in integration involves taking the limit of the Riemann sums as the number of subintervals increases indefinitely. This process ensures that the approximation converges to the actual area under the curve, which is the value of the definite integral, demonstrating the power of limits in bridging discrete approximations and continuous areas.

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