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Magazine Chapter 8: Problem 4
Evaluate the integrals without using tables. $$\int_{0}^{4} \frac{d x}{\sqrt{4-x}}$$
Short Answer
Expert verified
The integral evaluates to 4.
Step by step solution
01
Identify Substitution
Notice that the integrand contains the expression \( \sqrt{4-x} \). To simplify this, let's use a trigonometric substitution. We know that \( \sin^2 \theta = 1 - \cos^2 \theta \) and this can help us simplify the square root expression. We choose the substitution \( x = 4 - u^2 \) or \( u = \sqrt{4-x} \). This results in \( du = -\frac{1}{2\sqrt{4-x}} dx \). We will use this to substitute for \( dx \).
02
Adjust Limits
Change the limits of integration according to the substitution. When \( x = 0 \), \( u = \sqrt{4-0} = 2 \). When \( x = 4 \), \( u = \sqrt{4-4} = 0 \). The new limits for \( u \) will be from 2 to 0.
03
Substitute and Simplify
Based on our substitution \( x = 4 - u^2 \), and derivatives, rewrite the integral in terms of \( u \):\[ \int_{2}^{0} \frac{-2u \, du}{u} = \int_{2}^{0} -2 \, du \ = \int_{0}^{2} 2 \, du \]. The negative sign changes the limits so the integral can be rewritten from 0 to 2.
04
Integrate
Now integrate the function:\( \int_{0}^{2} 2 \, du = 2[u]_{0}^{2} \).Calculate the definite integral by applying the limits:\( 2[2 - 0] = 4 \).
05
Conclusion
The original integral \( \int_{0}^{4} \frac{dx}{\sqrt{4-x}} \) evaluates to 4 after substitution and integration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
A definite integral is used to calculate the accumulated quantity, like area under a curve, within a certain interval. It provides the precise value from one specific limit to another. In the exercise given, we evaluate a definite integral from 0 to 4:
- The expression inside the integral, \( \frac{dx}{\sqrt{4-x}} \), represents the function whose area under the curve we are interested in.
- The limits 0 and 4 signify the start and the end of integration, making the solution only applicable to this particular range.
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals, particularly those involving square roots. This method replaces variables with trigonometric functions, taking advantage of trigonometric identities.
- For instance, \( \sqrt{1-\sin^2 \theta} = \cos \theta \), so substituting \( x = 4 - u^2 \) is analogous to using a sine function since \( \sin^2 \theta = 1 - \cos^2 \theta \).
- In our problem, the choice of substitution \( x = 4 - u^2 \) helps simplify the square root to a form easier to integrate.
Change of Limits
Changing the limits of integration is a crucial step in the process of substitution in definite integrals. When you substitute a variable in, you must convert the original limits from the initial variable to the new one.
- In this exercise, when \( x = 0 \), substitute to find \( u = \sqrt{4-0} = 2 \).
- Conversely, when \( x = 4 \), \( u = \sqrt{4-4} = 0 \).
- Hence, the original limits from 0 to 4 for \( x \) are adjusted to 2 to 0 for \( u \).
Definite Integral Evaluation
Evaluating a definite integral involves performing the integration and then applying the limits of integration to find the actual value.
- In the exercise, after substituting and simplifying, we encounter \( \int_{0}^{2} 2 \, du \).
- The integration of \( 2 \, du \) gives \( 2u \), a simple antiderivative.
- Applying the definite integral limits results in calculating \( 2[2 - 0] \), leading to the final answer, 4.
