The integral ∫1∞ 1 x2 dx evaluates to a finite area under the curve from 1 to infinity, concluding the journey with a destination. Therefore, it is convergent.
The integral, ∫1∞ 1 x2 dx, can be interpreted as an adventure through the realm of calculus, much like exploring a vast landscape through literature. To determine the nature of this journey, namely whether it is convergent or divergent, we must assess whether the total area under the curve of f(x) = 1/x^2 from 1 to infinity is finite or infinite. The function 1/x^2 is a decreasing function, meaning as x increases, the function values get smaller and approach zero. It can be seen as a series of infinitesimally small strips whose areas we sum over this journey, as depicted in Figure 7.8.
Upon evaluating the integral, we find that the antiderivative of 1/x^2 is -1/x. Plugging in the bounds from 1 to infinity, we have the limit of -1/x as x approaches infinity which is 0, minus the antiderivative evaluated at the lower bound, which is -1. This yields a value of 1. Hence, the journey has a destination; the total area under the curve from 1 to infinity is finite, and our integral represents a captivating novel with an ending; it is convergent.
The probable question may be:
Imagine you are exploring a vast library of books, each representing a unique mathematical concept. One of the books has an intriguing question about an integral. The integral in question is:
∫1∞1x^2 dx
As a humanities enthusiast, your task is to determine whether this mathematical expression is like a captivating novel with an ending (convergent) or more like an open-ended exploration (divergent). Please choose the appropriate option:
Convergent
Divergent
Additional Information:
Consider this integral as a journey from 1 to infinity, exploring the mathematical landscape as you go. The integrand, 1/x^2, represents a reciprocal relationship where the farther you travel, the smaller the values become. Think of it as an adventure into the mathematical realm, and decide whether this particular journey reaches a destination (converges) or continues indefinitely (diverges).