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Chapter 5: Problem 6

Evaluate the integral. $$ \int_{-6}^{-1} 8 d x $$

Short Answer

Expert verified
The value of the integral is 40.

Step by step solution

01

Identify the integral format

The given integral is a definite integral of the form \( \int_{a}^{b} c\, dx \), where \(c\) is a constant. Here, \(c = 8\), \(a = -6\), and \(b = -1\).
02

Apply the integration rule for constants

The integral of a constant \(c\) with respect to \(x\) over a range is \(c \cdot (b - a)\). Thus, the indefinite integral for this constant \(8\) is \(8x\).
03

Plug in the limits of integration

Using the fundamental theorem of calculus, evaluate the indefinite integral at the upper limit \(b = -1\) and lower limit \(a = -6\), then take the difference:\[[8x]_{-6}^{-1} = 8(-1) - 8(-6)\]
04

Calculate the definite integral

Substitute the bounds into the expression:\[8(-1) - 8(-6) = -8 + 48 \]Calculate the result: \[-8 + 48 = 40\].

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Key Concepts

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

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a cornerstone of calculus that bridges the concepts of differentiation and integration. It shows how an antiderivative of a function can be used to evaluate a definite integral. This theorem asserts that if a function is continuous on a closed interval \([a, b]\), then the integral of the function over \([a, b]\) can be determined using the antiderivative:
  • Evaluate the antiderivative (or indefinite integral) at the upper limit of integration, then
  • Evaluate the antiderivative at the lower limit of integration.
  • Subtract the second result from the first.
This process can be compactly described as \([F(b) - F(a)]\), where \([F(x)]\) is the antiderivative.
Constant Integration
In calculus, when integrating a constant function, the process is relatively simple. If you have a constant \(c\) and you wish to integrate it with respect to \(x\), the result is straightforward:
  • The integral of a constant \(c\) over a variable \(x\) is given by \(c \cdot x\).
  • When dealing with definite integrals, this result must then be applied to the limits of integration.
For example, in the integral \(\int_{-6}^{-1} 8 \, dx\), you integrate to get \(8x\). This represents how this function accumulates from the lower to the upper limit.
Limits of Integration
The limits of integration are essential in the evaluation of definite integrals. These boundaries \([a, b]\) determine the interval over which the area under the curve is calculated. Here is how they work:
  • The lower limit \(a\) specifies where the accumulation of area starts.
  • The upper limit \(b\) indicates where the area accumulation ends.
  • In determining the definite integral \(\int_{a}^{b} f(x)\, dx\), you compute \(F(b) - F(a)\).
By plugging these values into the antiderivative, you extract the total accumulation of the function over the specified interval. For \(\int_{-6}^{-1} 8 \, dx\), it results in calculating \(8(-1) - 8(-6)\) to find the accumulated total.

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