Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$

Step-by-Step Solution

Answer

The improper integral converges if both partitioned integrals converge. The value is then determined by summing up the evaluations of both integrals.

Step 1: Partition the Integral

Partition the integral at the point of discontinuity - the point where the denominator equals zero, i.e., \(x = 1\). This results in two separate integrals: \[ \int_{0}^{1} \frac{1}{\sqrt[3]{x-1}} dx \] and \[ \int_{1}^{2} \frac{1}{\sqrt[3]{x-1}} dx \]

Step 2: Determine Convergence or Divergence

Using the limit as \(x\) approaches from the left for the first integral and from the right for the second integral, if both integrals converge, then the original integral also converges, otherwise it diverges. The antiderivative of \(1/\sqrt[3]{x-1}\) is \(3/2 \cdot (x-1)^{2/3}\). This can be applied to both integrals and evaluated at their respective endpoints.

Step 3: Evaluate the Integrals

Evaluating the integrals and taking the sum of the results from both, if both integrals converged, provides the value of the original integral.