Question

The inequality sec \(x \geq 1 + \left( x ^ { 2 } / 2 \right)\) holds on \(( - \pi / 2 , \pi / 2 ) .\) Use it to find a lower bound for the value of \(\int _ { 0 } ^ { 1 } \sec x d x .\)

Short answer

The value of the integral \(\int _ { 0 } ^ { 1 } \sec x \, dx\) is greater than or equal to 7/6

Step-by-step solution

Re-arrange the inequality

Start with the inequality \(sec x \geq 1 + \left( x ^ { 2 } / 2 \right)\). Re-arrange the inequality such that sec \(x\) - 1 is on the left side and \(x ^ { 2 } / 2\) on the right side like this: \(sec x - 1 \geq x ^ { 2 } / 2\)

Verify the domain

It is given that the inequality holds on the interval \( (-\pi/2 , \pi/2)\). This domain is the full range where sec \(x\) is defined.

Determine the lower bound of the integral

Now we need to find a lower bound for the value of \(\int _ { 0 } ^ { 1 } \sec x \, dx \). Replace sec \(x\) in the integral with \(1 + \left( x ^ { 2 } / 2 \right)\) (the lower bound found in Step 1) to get \(\int _ { 0 } ^ { 1 } \left(1 + \frac{x^2}{2}\right) \, dx\)

Compute the Integral

Now compute the integral, this can be split into two parts due to linearity of the integral, yielding the integral of 1 from 0 to 1 plus the integral of \(x^2/2\) from 0 to 1: \(\int ^ { 1 } _ { 0 } dx + \frac{1}{2}\int _ { 0 } ^ { 1 }x^2 \, dx = [x]_0^1 + \frac{1}{2}[x^3/3]_0^1 = 1 + \frac{1}{2}*(1/3) = 1 + 1/6 = 7/6\

Key concepts

Trigonometric Inequality

Understanding trigonometric inequalities is crucial when dealing with integration problems that involve trigonometric functions. These inequalities provide bounds that can simplify the integration process. In our exercise, we're given a specific inequality, \( sec x \text{\geq} 1 + \left( x^2 / 2 \right) \). This tells us that for all values of \( x \) within the interval \( (-\pi/2, \pi/2) \), \( sec x \) is always greater than or equal to \(1 + (x^2/2)\).

Knowing this relationship is important because it can be used to establish a minimum value, or lower bound, for a given function within a specified domain—which, in our case, is necessary to estimate the lower bound of an integral involving the secant function. It's key to note that such inequalities rely on the domain; if the domain changes, the inequality may not hold true.

Definite Integral

The definite integral is a fundamental concept in calculus, representing the accumulation of quantities and the area under a curve. It is defined over a specific interval, in our case \([0, 1]\), and is distinct from the indefinite integral, which represents an antiderivative and includes a constant of integration. A definite integral is notated as \(\int _ { a }^{ b } f(x) dx\), which computes the net area from \(x=a\) to \(x=b\) under the curve defined by the function \(f(x)\).

In practical terms, the definite integral tells us the total 'amount' of whatever the function \(f(x)\) represents, between two points. When applying definite integrals to real-world problems, this could mean anything from the total distance traveled over time to the accumulated profit over a range of sales volume.

Lower Bound Estimation

Lower bound estimation is a technique often employed in numerical analysis and calculus to approximate the minimum value of an integral. The idea is to find a function that is always less than or equal to the function being integrated over the interval of interest. In our case, using the trigonometric inequality given, we've determined that \(1 + (x^2/2)\) serves as a lower bound for \(sec x\) on the interval \((-\textbackslash pi/2, \textbackslash pi/2)\).

By using the lower bounding function in the integral \(\int _ { 0 } ^ { 1 } sec x dx\), we're able to estimate a number that the true value of the integral definitely surpasses, ensuring that we do not overestimate the integral's value. This is a conservative approach to integral estimation that can prove useful in a variety of scenarios, such as risk assessment and quality control.

Integral Computation

Integral computation involves finding the value of definite or indefinite integrals. In this step-by-step solution, the integral computation is broken down into simpler parts using the linearity property of integrals. The integral of a sum can be divided into the sum of the integrals—the integral of the constant 1 and the integral of the term \(x^2/2\) are computed separately.

Specifically, the computation results in \(\int ^ { 1 } _ { 0 } dx = [x]_0^1\), which evaluates to 1 since it's the integral of a constant over an interval of length 1, and \(\frac{1}{2}\int _ { 0 } ^ { 1 } x^2 dx = \frac{1}{2}[\frac{x^3}{3}]_0^1\), which evaluates to \(1/6\) after simplifying. By summing these results, we find that the lower bound for our integral is \(1 + 1/6 = 7/6\), which means the actual value of the definite integral \(\int _ { 0 }^ { 1 } sec x dx\) is at least 7/6.