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Chapter 5: Problem 17

Evaluate the integral. $$ \int_{1}^{1}\left(4 x^{2}-5\right)^{100} d x $$

Short Answer

Expert verified
The integral evaluates to 0.

Step by step solution

01

Identify the Integral Limits

The integral is given as \[\int_{1}^{1}\left(4 x^{2}-5\right)^{100} \, dx\]The limits of the integral are both 1, meaning the lower and upper limits of integration are the same.
02

Apply the Property of Definite Integrals

One of the properties of definite integrals is that if the limits of integration are the same (i.e., \(a = b\)), the value of the integral is zero:\[\int_{a}^{a} f(x) \, dx = 0\]In our case, \(a = 1\) and \(b = 1\), so the integral becomes zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Definite Integrals
When it comes to definite integrals, understanding their properties is essential. One key property is their behavior when the limits of integration are the same. When you see the notation \(\int_{a}^{a} f(x) \, dx\), where the lower and upper limits are equal, the integral evaluates to zero. This is because the integral represents the "net area" under the curve \(f(x)\) from \(a\) to \(b\).
If \(a\) equals \(b\), there is no interval to calculate the area over, resulting in a value of zero.
  • This property is immensely useful for quickly identifying when an integral can be evaluated instantly.
  • It saves time and cognitive load, allowing for a more efficient problem-solving process.
  • Understanding this can often simplify complex problems with minimal effort.
Integral Limits
Integral limits, denoted by \(a\) and \(b\) in \(\int_{a}^{b} f(x) \, dx\), define the interval over which a function is integrated. They are crucial as they determine the start and end points of the calculation of the "net area" under the curve.
  • If the limits are the same鈥攎eaning \(a = b\)鈥攊t indicates there is no width to the interval, leading directly to an integral value of zero.
  • This behavior stems from the fact that integration is essentially the process of summing infinitesimal slices of an area. Without a distance to sum over, the total sum is zero.
When working with integrals, carefully identifying your limits can often reveal valuable insights or simplify the problem at hand. It is always helpful to start with clearly understanding this basic aspect of definite integrals.
Calculus Problem-Solving
Solving calculus problems, especially with integrals, involves a strategic approach. First, identify the integral function and understand its boundaries, \(a\) and \(b\). This simple step already gives a lot of information about the nature of the problem.
  • If you notice the limits are the same, as in \(\int_{1}^{1} f(x) \, dx\), the problem simplifies immediately due to the properties of definite integrals.
  • Such recognition saves time and demonstrates a deep understanding of calculus concepts.
Effective problem-solving in calculus combines recognizing patterns, like this integral property, with a solid foundation of general calculus principles.
Remember, sometimes the problem isn't about complex calculations but spotting simple, elegant solutions. Always keep a lookout for basic properties that simplify your work, enhancing both accuracy and speed.

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