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Magazine Chapter 5: Problem 24
In Exercises 23-34, evaluate the definite integral. $$\int_{0}^{\sqrt{2}} \frac{1}{\sqrt{4-x^{2}}} d x$$
Short Answer
Expert verified
\(\frac{\pi}{4}\)
Step by step solution
01
Choose appropriate substitution
In order to simplify the integral, a trigonometric substitution is chosen. Here, \(x = 2 \sin(\theta)\) is a good choice. This makes \(dx = 2 \cos(\theta) d \theta\) and simplifies our denominator as \(\sqrt{4 - x^2} = \sqrt{4 - 4\sin^2(\theta)} = 2\cos(\theta)\)
02
Substitute the values into the integral
Substituting the previous results into the integral, we get: \(\int_{0}^{\sqrt{2}} \frac{1}{\sqrt{4-x^{2}}} dx = \int \frac{2 \cos (\theta)}{2 \cos (\theta)} d \theta = \int d\theta\)
03
Change the limits of the integral
Since the variable of integration has changed, the limits of the integral need to be changed as well. For \(x = 0\), \(\sin(\theta) = 0\) implies \(\theta = 0\). For \(x = \sqrt{2}\), \(\sin(\theta) = \frac{\sqrt{2}}{2}\) implies \(\theta = \frac{\pi}{4}\). Hence, the integral changes to \(\int_{0}^{\frac{\pi}{4}} d\theta\)
04
Evaluate the integral
\(\int_{0}^{\frac{\pi}{4}} d\theta = \left[\theta\right]_{0}^{\frac{\pi}{4}} = \frac{\pi}{4} - 0 = \frac{\pi}{4}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Substitution
Trigonometric substitution is a powerful technique used to simplify integrals, especially those involving square roots. This method involves substituting a trigonometric expression for a variable, which can make the integral easier to handle. In the exercise provided, we used the substitution \( x = 2 \sin(\theta) \). This choice is effective because it simplifies the form of \( \sqrt{4 - x^2} \) to \( 2\cos(\theta) \).
To apply trigonometric substitution, consider the following steps:
To apply trigonometric substitution, consider the following steps:
- Identify the right substitution based on the expression under the square root. For \( \sqrt{a^2 - x^2} \), use \( x = a \sin(\theta) \).
- Compute the differential \( dx \) after substituting, which usually involves finding \( dx = \frac{d}{d\theta}[\text{your substitution}]\, d \theta \).
- Simplify the integral using the trigonometric identity, here \( \sin^2(\theta) + \cos^2(\theta) = 1 \), to make the expression more manageable.
Integration Limits
Changing the limits of an integral is necessary when performing a substitution because the variable has changed. In our example, we originally had limits in terms of \( x \), going from 0 to \( \sqrt{2} \). After substitution with \( x = 2\sin(\theta) \), we need new integration limits.
Here鈥檚 how you update the limits:
Here鈥檚 how you update the limits:
- Substitute the original lower limit into the substitution equation to find the new lower limit. For \( x = 0 \), we find \( \sin(\theta) = 0 \), thus \( \theta = 0 \).
- Substitute the original upper limit into the substitution to find the new upper limit. For \( x = \sqrt{2} \), we find \( \sin(\theta) = \frac{\sqrt{2}}{2} \), thus \( \theta = \frac{\pi}{4} \).
Integral Evaluation
Evaluating the integral after substitution is simplified once the limits and the integrand are transformed correctly. For our transformed integral, we had \( \int_{0}^{\frac{\pi}{4}} d\theta \). This is a straightforward integral because \( d\theta \) indicates we're integrating \( 1 \times d\theta \).
The process includes:
The process includes:
- Identify the resultant expression after substitution, which in this case simplified to integrating just \( d\theta \).
- Perform the integration. The integral of \( 1 \) with respect to \( \theta \) is \( \theta \).
- Apply the new limits: Substitute the limits back into the integrated function \( \theta \).
Calculus
Calculus is a branch of mathematics that studies how things change. It provides a framework for modeling systems where change is a prominent feature. When we evaluate a definite integral, we're essentially calculating the accumulated change over a specific interval.
There are a few fundamental ideas to grasp to master calculus:
There are a few fundamental ideas to grasp to master calculus:
- Understand the concept of limits, which lay the foundation for the derivative and integral.
- The idea of differentiation 鈥� finding out how a function changes at any given point.
- Integration, which involves finding the total change over an interval by computing the area under a curve.
