To sketch the region bounded by the graphs of the given algebraic functions, we first need to plot the graphs of these functions on a coordinate plane.
Let's start by graphing the function f(x) = 10/x. Since this function has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0, it will have a shape similar to a hyperbola.
To plot the graph of f(x), choose some values of x and calculate the corresponding y values. For example, let's use x = -1, 1, 2, 5, and 10. Evaluating f(x) for these x-values, we get:
f(-1) = -10
f(1) = 10
f(2) = 5
f(5) = 2
f(10) = 1
Plotting these points on a graph, we can see that the graph of f(x) passes through the points (-1, -10), (1, 10), (2, 5), (5, 2), and (10, 1). Draw a smooth curve that connects these points.
Now, let's find the region bounded by the graphs of f(x), x = 0, y = 2, and y = 10. Looking at the graph, we can see that the region is enclosed by the x-axis (x = 0), the vertical line x = 0, the horizontal line y = 2, and the graph of f(x).
To find the area of this region, we need to calculate the definite integral of f(x) with respect to x, between the appropriate limits.
Since the region is bounded by the vertical line x = 0 and the graph of f(x), we need to find the definite integral of f(x) from x = 1 to x = 10 (as it is bounded by the points (1, 10) and (10, 1)).
Using the definite integral formula, the area of the region is given by:
Area = ∫(from x=1 to x=10) [f(x)] dx
Substituting the equation for f(x), we get:
Area = ∫(from x=1 to x=10) (10/x) dx
To integrate this function, we can use the logarithmic property of integration. So, the area can be calculated as:
Area = [10ln|x|] (from x=1 to x=10)
Evaluating this expression for the limits, we get:
Area = [10ln|10|] - [10ln|1|]
= 10ln(10) - 0
= 10ln(10)
Therefore, the area of the region bounded by the graphs of the functions f(x), x = 0, y = 2, and y = 10 is 10ln(10) square units.