91Ó°ÊÓ

In calculus, definite integrals are used to calculate the area under a curve within a specified interval. Unlike indefinite integrals, which represent a family of functions, definite integrals provide a specific numerical value. They are expressed as \(\text{∫}_{a}^{b} f(x) \text{d}x\). The integral computes the accumulation of quantities, such as area, over the interval \[a, b\].
For example, in our problem, to find the area under the curve \(y=\frac{1}{x^2}\) from \(x=1\) to \(x=\text{∞}\), we set up the integral \(\text{∫}_{1}^{∞} \frac{1}{x^2} \text{d}x\).
Here’s a simple step-by-step method to solve a definite integral:

• Determine the function \(f(x)\) you need to integrate.
• Identify the interval \[a, b\] over which to integrate.
• Find the antiderivative (indefinite integral) of \(f(x)\).
• Evaluate the antiderivative at the boundaries \((b \text{ and } a)\).
• Subtract the value at \(a\) from that at \(b\) to get the area.

Calculating the area under a curve is a fundamental application of definite integrals. The area represents a physical or graphical idea, like the amount of paint needed to cover a surface. For functions like \(y = \frac{1}{x^2}\), this calculation determines whether you can successfully paint the specified region.
When we work with the function \(y = \frac{1}{x^2} \text{ from } x=1 \text{ to } x=\text{∞}\), the corresponding integral \(\text{∫}_{1}^{∞} \frac{1}{x^2} \text{d}x\) equals 1, showing that the area is finite. However, for \(y=\frac{1}{x}\), the integral \(\text{∫}_{1}^{∞} \frac{1}{x} \text{d}x\) diverges to infinity, indicating an infinite area.
In practical terms:

• Finite integral: The area under \(y=\frac{1}{x^2}\) could be covered with a finite amount of paint.
• Infinite integral: The area under \(y=\frac{1}{x}\) would require an endless amount of paint, making it impossible to cover fully.

Improper integrals often arise when dealing with infinite intervals, as seen in our problem where the integration spans from \(x=1\) to \(x=\text{∞}\). Such integrals need careful evaluation to determine if they converge (result in a finite area) or diverge (result in an infinite area).
For example, to determine whether the region under \(y=\frac{1}{x^2}\) is finite over \[1, \text{∞}\], we calculate: \[ \text{∫}_{1}^{∞} \frac{1}{x^2} \text{d}x = \bigg[ -\frac{1}{x} \bigg]_{1}^{∞} = (0 - (-1)) = 1. \] This shows a finite area of 1.
Conversely, for \(y=\frac{1}{x}\), the integral: \[ \text{∫}_{1}^{∞} \frac{1}{x} \text{d}x = \bigg[ \text{ln}|x| \bigg]_{1}^{∞} = (\text{∞} - \text{ln}(1)) = \text{∞} \] diverges, indicating an infinite area.
To handle infinite intervals:

• Transform the integral limits to capture the behavior as they approach infinity.
• Evaluate to see if the integral converges or diverges.
• For convergence, you get a finite area; for divergence, the area is infinite.