Problem 20 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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: \( \int_1^{+\infty}|x-1 - \sqrt[3]{x-1}| dx \)

Step 4: 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.