Q16Mathematics I

The mathematical challenge is to accurately evaluate the double integral:

An immediate attempt to integrate this directly with respect to $y$ fails entirely, because the antiderivative of $1/\log y$ (the logarithmic integral function) cannot be expressed in terms of elementary mathematical functions. Therefore, changing the order of integration to $dxdy$ is absolutely mandatory.

Step 1: Sketching the Original Region

The original limits of integration strictly define the bounded 2D region. - The inner limits dictate that $y$ bounds vertically from the exponential curve $y=e^x$ up to the horizontal constant line $y=e$. - The outer limits dictate that $x$ bounds horizontally from $x=0$ to $x=1$. This creates a curvilinear triangular region in the first quadrant, bounded entirely by the y-axis, the line $y=e$, and the exponential curve.

Step 2: Defining the New Limits

To change the order to $dxdy$, we must redefine the same region using horizontal sweeping strips instead of vertical ones. - Outer limits (for $y$): The absolute lowest point of this physical region occurs at $x=0$, where $y=e^0=1$. The absolute highest point is the line $y=e$. Therefore, $y$ strictly varies from $1$ to $e$. - Inner limits (for $x$): A horizontal strip entering the region from the left always touches the y-axis first ($x=0$) and exits the region on the right by touching the curve $y=e^x$. To express this curve limit in terms of $y$, we mathematically invert the function: $x=\log y$. Thus, $x$ strictly varies from $0$ to $\log y$.

The newly structured integral becomes:

Step 3: Execution of the New Integral

The integration is now trivial. For the inner integral, we integrate strictly with respect to $x$. Since $\frac{1}{\log y}$ is entirely independent of $x$, it acts as a mere constant:

Evaluating the strict limits:

The complex logarithmic terms cancel out perfectly, leaving the integral of $1$:

Thus, the exact numerical evaluation of the integral is definitively $e-1$.