Problem 178 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int_{0}^{10 \pi}|\sin x| d x\) is (A) 20 (B) 8 (C) 10 (D) 18

Step-by-Step Solution

Answer

The answer is (A) 20.

Step 1: Understand the Problem

We need to evaluate the integral \( \int_{0}^{10\pi} |\sin x| \, dx \). The integrand is the absolute value of the sine function, which affects its graph and integration over intervals.

Step 2: Analyze the Function

The function \( \sin x \) has a period of \( 2\pi \). Within one period (\( 0 \) to \( 2\pi \)), \( |\sin x| \) is symmetric and covers both the 0 to \( \pi \) and \( \pi \) to \( 2\pi \) intervals equally.

Step 3: Simplify the Problem Using Symmetry

From 0 to \( \pi \), \( |\sin x| = \sin x \). From \( \pi \) to \( 2\pi \), \( |\sin x| = -\sin x \) which is the same as \( \sin x\) reflected in the x-axis. Thus, \( \int_0^{2\pi} |\sin x| dx = 2\int_0^{\pi} \sin x \, dx = 4 \).

Step 4: Break Down the Integral

Our range \( 0 \) to \( 10\pi \) covers 5 full periods of \( 2\pi \). Calculate \( 5 \times \int_0^{2\pi} |\sin x| dx = 5 \times 4 \) since the function repeats every \( 2\pi \).

Step 5: Solve the Integral

\( 5\times 4 = 20 \). This is the total area under \( |\sin x| \) from \( 0 \) to \( 10\pi \).

Step 6: Conclusion

Therefore, the value of the integral \( \int_{0}^{10\pi} |\sin x| \, dx \) is 20.

Key Concepts

absolute value function

The absolute value function is a unique mathematical concept that transforms numbers into their non-negative equivalents. When you see the notation \(|x|\), it simply means that you are taking the positive version of \(x\). This function alters the graph of other functions significantly. It flips any negative sections up into the positive region. This effect is crucial when integrating functions like sine, as it changes how we interpret the area under the curve.

• The graph of an absolute value function looks like a "V".
• It mirrors any negative values to the positive side.
• In the context of \(\sin x\), the negative parts from \(\pi\) to \(2\pi\) are flipped up.

Remember, the absolute value can dramatically change the behavior of a function, especially when calculating areas under curves with integration.

sine function

The sine function, represented as \(\sin x\), is a fundamental trigonometric function that describes smooth, wave-like oscillations. You'll often encounter it in various fields such as physics, engineering, and mathematics. Sine functions vary periodically, producing a wavy pattern that repeats every \(2\pi\) radians. This periodic behavior is pivotal in calculus when tackling integrations and transformations, like the absolute value.

• Sine naturally oscillates between -1 and 1.
• Completes one full cycle in \(2\pi\) radians.
• Common in modeling periodic phenomena like sound waves and light.

When integrating \(|\sin x|\), understanding the wave pattern of the original sine function is key to evaluating the integral correctly.

periodic functions

Periodic functions are functions that repeat their values at regular intervals, called periods. The sine function is an example of a periodic function with a regular interval of \(2\pi\). With periodic functions, once you understand the behavior over one period, you can predict the entire function across any range. This feature is handy when integrating over large intervals, as it simplifies the computations.

• A periodic function like \(\sin x\) repeats every \(2\pi\).
• This repeat allows you to break integrations into easier, smaller parts.
• Understanding one full cycle gives insight into the entire pattern.

For an integral like \(\int_{0}^{10\pi}|\sin x| dx\), recognizing that it covers multiple full periods helps determine the total area.

integration methods

Integration methods include various techniques and strategies to calculate the area under curves or to find the accumulation of quantities. Common methods include substitution, integration by parts, and symmetry exploits, especially for periodic functions. For periodic functions, such as the absolute value of the sine, symmetry becomes a powerful tool. By understanding the pattern over a single period, you can compute the entire integral simply by multiplying.

• Basic integration uses formulas or anti-derivatives.
• Symmetry reduces workload when dealing with periodic integrals.
• Multiplying the integral over one period by the number of periods simplifies lengthy evaluations.

In the exercise, integrating \(\int_{0}^{10\pi}|\sin x| dx\) uses symmetry across its periods to efficiently find the result.