Question

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(x)=\sqrt[3]{x-1}, g(x)=x-1 $$

Short answer

The area under the curve is undefined as it is infinite.

Step-by-step solution

Sketch the Functions

First, sketch both functions \( f(x)=\sqrt[3]{x-1}\) and \(g(x)=x-1\) on the same graph for an initial visual understanding of the region. The function \( f(x)=\sqrt[3]{x-1} \) is a cubic root function shifted 1 unit to the right, and \( g(x)=x-1 \) is a straight line with a slope of 1.

Find the Intersection Points

Set \(f(x) = g(x)\) to find the points where the graphs intersect. These points will be the limits of integration when finding the area. Thus, by setting \(\sqrt[3]{x-1} =x-1\) and solving the equation, obtain \(x=1\) as the point of intersection.

Set Up the Integral for Area Calculation

Now, set up the integral for calculating the area between the two curves. The formula for the area between two curves from a to b is \( \int_a^b |f(x) - g(x)| dx\). Since the number of intersection points is only one, another value for 'b' in the integral can't be found. But by observing the graphs, it can be concluded that \(g(x)=x-1\) overtakes \(f(x)=\sqrt[3]{x-1}\) as \( x \to +\infty \). Thus the integral setting for the calculation becomes: \( \int_1^{+\infty}|x-1 - \sqrt[3]{x-1}| dx \)

Calculate the Area

Finally, compute the integral, an improper one, to find the area bounded by these two curves. However, this integral doesn't converge to a finite area. Therefore, the concept of the 'area' between these two curves doesn't exist in the usual sense as the area under the curve is infinite.