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Chapter 6: Problem 38

Write as a single definite integral. \(\int_{0}^{8} f(x) d x+\int_{4}^{0} f(x) d x\)

Short Answer

Expert verified
\(\int_{0}^{8} f(x) \, dx\)

Step by step solution

01

Simplify the Definite Integral

In the given expression, notice that the second integral is defined from 4 to 0, which is in reverse order. According to the properties of definite integrals, \[ \int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx. \] This means we can rewrite\[ \int_{4}^{0} f(x) \, dx = -\int_{0}^{4} f(x) \, dx. \]
02

Combine the Integrals

Now, rewrite the expression using the property identified in Step 1:\[ \int_{0}^{8} f(x) \, dx + \int_{4}^{0} f(x) \, dx = \int_{0}^{8} f(x) \, dx - \int_{0}^{4} f(x) \, dx. \]We can combine these two integrals since they share the same lower limit:\[ = \int_{0}^{4} f(x) \, dx + \int_{4}^{8} f(x) \, dx. \]\[ = \int_{0}^{8} f(x) \, dx. \]

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Key Concepts

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

Integral Properties
When dealing with definite integrals, understanding their key properties can be incredibly useful in solving calculus problems efficiently. One important property is related to reversing the limits of integration.
  • If the limits of a definite integral are swapped, the integral's value changes its sign. This is expressed mathematically as \( \int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx \).
In our problem, we deal with an integral from 4 to 0, \( \int_{4}^{0} f(x) \, dx \), which can be rewritten as \(-\int_{0}^{4} f(x) \, dx \) according to this property. This reversal technique simplifies complex integral expressions and can make larger integrations possible.
Integral Simplification
Simplifying integrals can make solving calculus problems more manageable. Once you have understood the properties, such as reversing limits, you can combine or separate integrals based on their limits and functions.An initial expression in this problem includes two integrals with overlapping limits: \( \int_{0}^{8} f(x) \, dx + \int_{4}^{0} f(x) \, dx \). After using the property of reversing limits, the expression changes to:
  • \( \int_{0}^{8} f(x) \, dx - \int_{0}^{4} f(x) \, dx \).
When these integrals share a common limit (in this case, 0 to 4), they can be combined or rearranged. This results in a single integral from 0 to 8, \( \int_{0}^{8} f(x) \, dx \), which is far simpler to evaluate or understand.
Calculus Problem Solving
Solving calculus problems often involves a step-by-step approach, using properties and simplification techniques to reach a solution. In this case, we used definite integral properties and simplification methods to solve a seemingly complex integral.By rewriting the original exercise using known properties and simplification strategies, two separate integrals were combined into one. The process involved:
  • Identifying integrals in reverse limits and changing their signs.
  • Rewriting and combining the integrals sharing common limits.
  • Simplifying the expression into a single definite integral \( \int_{0}^{8} f(x) \, dx \).
This workflow is typical in calculus problem-solving. Learning how to identify opportunities for simplification can greatly ease the problem-solving process, leading to quicker and more accurate solutions.

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Most popular questions from this chapter

Find the producers' surplus, using the given supply equations and the equilibrium price \(p_{0}\). $$ S(x)=e^{x}, p_{0}=e^{2} $$

Suppose copper is being extracted from a certain mine at a rate given by \(\frac{d}{d t} P(t)=100 e^{-0.2 t},\) where \(P(t)\) is measured in tons of copper and \(t\) is measured in years from opening the mine. At the beginning of the sixth year a new mining innovation is introduced to boost production to a rate given by \(Q^{\prime}(t)=500 / t .\) Find the increase in production of copper due to this innovation during the second 5 years of its use over what copper production would have been without its use.

You are given the demand and supply equation. Find the equilibrium point, and then calculate both the consumers' surplus and the producers' surplus. $$ D(x)=50-x^{2} \text { and } S(x)=5 x $$

Use the order properties of the definite integral to establish the inequalities. $$ 24 \leq \int_{-4}^{4} \sqrt{9+x^{2}} d x \leq 40 $$

In Exercises 1 through 38 , find the antiderivatives. $$ \int\left(e^{x}-3 x\right) d x $$
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