Plot the graph of y=x^3 from x=0 to 1 . i Use the leftbox...

Plot the graph of $y=x^{3}$ from $x=0$ to 1 . i Use the leftbox command to draw four boxes on the graph of $y=x^{3}$. Evaluate the total area of the four boxes by using the leftsum command. ii Repeat $\mathbf{i}$ for 8 boxes. iii Repeat $\mathbf{i}$ for 16 boxes. iv Repeat $\mathbf{i}$ for 32 boxes. v Repeat $\mathbf{i}$ for 64 boxes. vi Repeat $\mathbf{i}$ for 128 boxes. vii By applying the command $\operatorname{evalf}\left(\operatorname{int}\left(x^{\wedge} 3, x=0.1\right)\right)$, evaluate $\int_{0}^{1} x^{3} \mathrm{~d} x .$ What do you notice about your results?

Plot the graph of $y=x^{3}$ from $x=0$ to 1 .
i Use the leftbox command to draw four boxes on the graph of $y=x^{3}$. Evaluate the total area of the four boxes by using the leftsum command.
ii Repeat $\mathbf{i}$ for 8 boxes.
iii Repeat $\mathbf{i}$ for 16 boxes.
iv Repeat $\mathbf{i}$ for 32 boxes.
v Repeat $\mathbf{i}$ for 64 boxes.
vi Repeat $\mathbf{i}$ for 128 boxes.
vii By applying the command $\operatorname{evalf}\left(\operatorname{int}\left(x^{\wedge} 3, x=0.1\right)\right)$, evaluate $\int_{0}^{1} x^{3} \mathrm{~d} x .$ What do you notice about your results?

Key Concepts

Convergence of Numerical Approximations

As the number of subdivisions (or rectangles) increases in a numerical integration method like the Left Riemann Sum, the approximation of the integral becomes more accurate. This phenomenon of the approximation tending towards an exact limit as the interval becomes increasingly refined demonstrates the principle of convergence, which is central to numerical analysis and calculus.

Definite Integral

A definite integral represents the net area under a curve between two specified endpoints. In the context of calculus, it is the limit of a sum of areas of shapes (often rectangles) that approximate the region under a function's graph, as the number of shapes approaches infinity. It provides an exact value for the accumulation of quantities, such as area, if the function is continuous over the interval.

Riemann Sum

A Riemann sum is a method used to approximate the value of a definite integral by summing the areas of a finite number of rectangles whose heights are determined by the value of the function at specific sample points. As the number of rectangles increases (with corresponding decreases in width), the approximation becomes more accurate, converging to the actual integral value.

Left Riemann Sum

The Left Riemann Sum is a specific type of Riemann sum where the function value at the left endpoint of each subinterval is used to determine the height of the rectangle. This method is effective for approximating the area under a curve, and as the subintervals are made smaller by increasing their number, the approximation generally improves and converges to the true value of the definite integral.

Graphing Functions

Graphing functions, such as plotting y = x³, is fundamental in understanding their behavior over an interval. Visualizing the function helps in comprehending the regions where the area is being calculated and aids in the selection of appropriate methods and number of divisions for the approximation process.

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