Problem 152 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The absolute value of \(\int_{10}^{19} \frac{\sin x}{1+x^{8}} d x\) is (A) less than \(10^{-7}\) (B) more than \(10^{-7}\) (C) less than \(10^{-6}\) (D) more than \(10^{-6}\)

Step-by-Step Solution

Answer

(A) less than \(10^{-7}\).

Step 1: Understand the Problem

We need to find an estimate of the absolute value of the integral \( \int_{10}^{19} \frac{\sin x}{1+x^{8}} dx \) and determine if it falls into one of the given ranges of magnitudes.

Step 2: Estimate the integral

We know that \( |\sin x| \leq 1 \). So, the integral can be estimated by:\[\left| \int_{10}^{19} \frac{\sin x}{1+x^{8}} dx \right| \leq \int_{10}^{19} \left| \frac{\sin x}{1+x^{8}} \right| dx \leq \int_{10}^{19} \frac{1}{1+x^{8}} dx\]

Step 3: Simplify the Integral

Focus on \( \int_{10}^{19} \frac{1}{1+x^{8}} dx \). The function \( \frac{1}{1+x^{8}} \) is very small since \( x^{8} \) grows very fast with \( x \geq 10 \).To approximate:The integral simplified can be expressed as:\[\int_{10}^{19} \frac{1}{x^{8}} dx\]

Step 4: Evaluate the Simplified Integral

Calculate \( \int \frac{1}{x^{8}} dx \):\[\int \frac{1}{x^{8}} dx = -\frac{1}{7x^7}\]Evaluate the definite integral from 10 to 19:\[\left[-\frac{1}{7x^7}\right]_{10}^{19} = \left(-\frac{1}{7(19)^7}\right) - \left(-\frac{1}{7(10)^7}\right)\]

Step 5: Calculate the Result

Compute the values:- \( 19^7 \approx 893871739 \) so \( \frac{1}{7\times 893871739} \approx 1.59 \times 10^{-10} \)- \( 10^7 = 10000000 \) so \( \frac{1}{7\times 10000000} \approx 1.4286 \times 10^{-8} \)The result is:\[1.4286 \times 10^{-8} - 1.59 \times 10^{-10} \approx 1.4127 \times 10^{-8}\]

Step 6: Determine the Category

Since \( 1.4127 \times 10^{-8} \) is clearly less than \( 10^{-7} \) and \( 10^{-6} \), the absolute value of the original integral is less than both thresholds, meaning option (A) is correct.

Key Concepts

Definite Integral

In calculus, a definite integral is a way to calculate the area under a curve, represented by a function, within a given interval on the x-axis. It is noted by \[ \int_{a}^{b} f(x) \, dx \] where

• a and b are the limits of integration,
• f(x) is the function to be integrated,
• dx indicates integration with respect to x.

The process involves evaluating the antiderivative of the function at these two points and finding their difference. This gives you the net area between the curve and the x-axis from a to b. If the function crosses the x-axis, the areas above and below the axis might cancel out since areas above the axis are positive, while those below are negative. Understanding definite integrals is crucial when you want to compute values such as the total displacement of an object from its velocity or the total accumulation of a quantity. In the given exercise, calculating the definite integral provides an estimate of the net area for the given function between the limits 10 and 19.

Absolute Value

The concept of absolute value in mathematics measures the magnitude of a number, irrespective of its sign. For any real number x, the absolute value is denoted as \( |x| \) and is defined as:

• \( x \) if \( x \geq 0 \)
• \(-x \) if \( x < 0 \)

This means the absolute value of a number is always non-negative.When dealing with definite integrals, finding the absolute value becomes important when you care about the size of the quantity rather than its direction. It ensures that any negative values within the integral's range are considered in terms of their magnitude. This is particularly important in real-world contexts such as physics and engineering where the direction of a force or quantity might not be relevant, only its size.In our exercise, determining the absolute value of the integral helps us understand how significant the impact of the function's variation is within the given limits. This gives a clearer picture of the magnitude of the quantity represented.

Estimation Techniques

Estimation techniques are essential mathematical tools that allow us to approximate complex or difficult-to-compute integrals and sums. These techniques help simplify calculations and provide insights into the magnitude and behavior of the function.One common estimation method is comparison. In the step-by-step solution, we estimate \[ \int_{10}^{19} \frac{\sin x}{1+x^8} \ dx \] by comparing it to a simpler function \[ \int_{10}^{19} \frac{1}{x^8} \ dx \]. Since \(|\sin x| \leq 1\), it provides a broader estimate. This simplified function is easier to integrate, allowing an easier approximation.A good estimate gives us a ballpark figure we can use to judge the size of the original integral without exactly calculating it. Estimation is important when exact solutions are time-consuming or impossible to find. By knowing how to bound or limit the integral between known values, it is possible to determine whether the final result fits within the specified categories, as was done in this problem to conclude that the integral is less than \(10^{-7}\) and \(10^{-6}\).