Problem 10 - Essential Calculus

Question

Explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. $$ \int_{2}^{2} \frac{-2}{(x-1)^{3}} d x=\frac{8}{9} $$

Step-by-Step Solution

Answer

The answer that the graphing utility provided, \(\frac{8}{9}\), is incorrect. The integral \(\int_{2}^{2} \frac{-2}{(x-1)^{3}} dx\) actually equals zero according to the properties of definite integrals. Therefore, the evaluated integral was incorrect.

Step 1: Evaluate the Integral Using Basic Properties

To begin the process, utilize a fundamental property of definite integrals: the integral from a to a of any function f(x) equals zero, written as: \[\int_{a}^{a} f(x) = 0 \]This is due to the fact that there is no area under the curve from one specific point to the same point. Hence even before computing the integral \( \int_{2}^{2} \frac{-2}{(x-1)^{3}} dx \), we can conclude that it equals zero.

Step 2: Compare the Integral's Value to the Graphing Utility’s Prediction

By using this definite integral property, we have established that \( \int_{2}^{2} \frac{-2}{(x-1)^{3}} dx = 0 \). Comparing this result with the answer given by the graphing utility (\(\frac{8}{9}\)) shows that the utility's answer is incorrect.

Step 3: Analyze the Error

The utility's incorrect answer usually indicates misunderstandings or coding errors involved in the calculation. In some cases, utilities can fail to account for the properties of definite integrals when programmed incorrectly.