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Chapter 5: Problem 11

In Exercises \(7-12,\) evaluate the integral. $$\int_{-2.1}^{3.4} 0.5 d s$$

Short Answer

Expert verified
The integral evaluates to \(2.75\).

Step by step solution

01

Identify the type of integral

The integral we are dealing with is a definite integral over a constant function. The basic theorem that states that the integral of a constant over the interval \([a, b]\) is equal to that constant multiplied by the size of the interval can be put to use here.
02

Calculate the size of the interval

Subtract the lower limit of integration from the upper limit: \(3.4 - (-2.1) = 5.5\). So, the size of the interval is \(5.5\).
03

Evaluate the integral

Finally, the value of the integral is equal to the constant (0.5 in this case) multiplied by the size of the interval. So the value of the integral will be \(0.5 * 5.5 = 2.75\).

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Key Concepts

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

Constant Function
When you encounter a constant function in calculus, you are dealing with a function where the output value is the same regardless of the input. In our exercise, the function is represented by the constant 0.5.

Since the function doesn't change, it simplifies the integration process significantly. Instead of worrying about curves or areas under varying graphs, you focus on simple multiplication. This constant output is a building block in calculus and provides a straightforward path for solving integrals involving constant functions.
Evaluation of Integrals
Evaluating an integral involves calculating the area under a curve defined by a function over a specified interval. In our example of a definite integral \[ \int_{-2.1}^{3.4} 0.5 \, ds \]it specifically tells us to find the area under the curve of the constant function 0.5 from -2.1 to 3.4.

Here鈥檚 how we evaluate it:
  • First, determine the size of the interval, which we find by subtracting the lower limit from the upper limit: \[ 3.4 - (-2.1) = 5.5 \]
  • Then, multiply the constant value by this interval length: \[0.5 \times 5.5 = 2.75 \]
This result, 2.75, tells you the exact quantity that represents the area under the constant curve within the specified range.
Properties of Definite Integrals
Understanding the properties of definite integrals can make solving problems faster and clearer. Here are a few essential properties:

  • **Linearity:** If you have \[\int_{a}^{b} c f(x) \, dx,\]where c is a constant, you can pull out the constant: \[c \times \int_{a}^{b} f(x) \, dx\]
  • **Additivity:** The integral from a to b can be split into the sum of two integrals: \[\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx \]
  • **Interval Length:** For a constant function, the length of the interval is crucial. As seen in our exercise, calculate this interval carefully to resolve accurate area under the curve. \[\text{If } f(x) = c, \text{ then } \int_{a}^{b} f(x) \, dx = c \times (b-a)\]
These properties help break down more complex integral problems into manageable pieces, allowing you to leverage basic numerical principles to find solutions.

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Most popular questions from this chapter

In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{2} x d x$$

Consider the integral \(\int_{-1}^{3}\left(x^{3}-2 x\right) d x\) (a) Use Simpson's Rule with \(n=4\) to approximate its value. (b) Find the exact value of the integral. What is the error, \(\left|E_{S}\right| ?\) (c) Explain how you could have predicted what you found in (b) from knowing the error-bound formula. (d) Writing to Learn Is it possible to make a general statement about using Simpson's Rule to approximate integrals of cubic polynomials? Explain.

In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { - 1 } ^ { 2 } 3 x ^ { 2 } d x$$

In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{-1}^{1} 2 \sqrt{1-x^{2}} d x$$

In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{2} x d x$$
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