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Chapter 12: Problem 33

Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1 $$

Short Answer

Expert verified
The area of the region is 1.

Step by step solution

01

Identify the region

The region is defined by the inequalities, \( y \leq \frac{1}{x^{2}} \), \( y \geq 0 \), \( x \geq 1 \). This is the region bounded by the curve \( y = \frac{1}{x^{2}} \) and the lines \( x = 1 \) and the x-axis.
02

Calculate the Definite Integral of the function

The area under the curve can be calculated by the definite integral of the function \( y = \frac{1}{x^{2}} \), from the lower limit of x which is 1, to the upper limit of x, which is infinity. So, the integral becomes \(\int_1^{\infty} \frac{1}{x^{2}} \, dx \).
03

Solve the Definite Integral

Solving the integral, which is \(\int_1^{\infty} \frac{1}{x^{2}} \, dx \), the integral of \( \frac{1}{x^{2}} \) is \( -\frac{1}{x} \). So, evaluate it at infinity and 1, we have \( -\frac{1}{\infty} + 1 \).
04

Get the Area

Using the property that any real number divided by infinity is zero, the final answer will be \( 0 + 1 = 1 \).

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Most popular questions from this chapter

$$ \int_{1 / 2}^{\infty} \frac{1}{\sqrt{2 x-1}} d x $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{2}\left(x^{4}+1\right) d x, n=4 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} 2 x e^{-3 x^{2}} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=2, n=4 $$
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