Chapter 26: Problem 3
Evaluate each definite integral to three significant digits. Check some by calculator. $$\int_{1}^{3} 7 x^{2} d x$$
Short Answer
Expert verified
60.7
Step by step solution
01
Apply the Power Rule for Integration
To evaluate the definite integral \(\int_{1}^{3} 7 x^{2} dx\), start by applying the power rule for integration, which states that the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\), plus a constant of integration C. For the integrand \(7x^2\), we integrate term by term to get \(\frac{7}{3}x^{3}\).
02
Perform the Definite Integration
After finding the indefinite integral, evaluate it between the limits of 1 and 3. This is done by computing \(\frac{7}{3}x^{3}\) at the upper limit (3) and subtracting the value of \(\frac{7}{3}x^{3}\) at the lower limit (1): \[ \left. \frac{7}{3}x^{3} \right|_1^3 = \frac{7}{3}(3^{3}) - \frac{7}{3}(1^{3}). \]
03
Calculate the Result
Now, calculate the result of the definite integral: \[ \frac{7}{3}(3^{3}) - \frac{7}{3}(1^{3}) = \frac{7}{3}(27-1) = \frac{7}{3}(26) = 7 \times 26 \times \frac{1}{3} = 7 \times 8.666 \dots = 60.666 \dots, \] to three significant digits, the result is 60.7.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule for Integration
When it comes to the world of integrals, the power rule for integration is your best friend. It simplifies the process of finding the indefinite integral of polynomial functions. Let's demystify this concept! The rule simply says that for a function of the form \( x^n \), where \( n \) is any real number except -1, the indefinite integral is \( \frac{1}{n+1}x^{n+1} \), plus a constant \( C \).
Think of it as giving power to the x variable! If we have \( x^2 \), we add 1 to the exponent to get \( x^3 \), and then divide by this new exponent, resulting in \( \frac{1}{3}x^3 \). Easy, right? This powerful rule works every time, making integration approachable and much less intimidating.
In our exercise, the power rule transformed the integrand \( 7x^2 \) into \( \frac{7}{3}x^3 \), setting the stage for finding the definite integral between specific limits.
Think of it as giving power to the x variable! If we have \( x^2 \), we add 1 to the exponent to get \( x^3 \), and then divide by this new exponent, resulting in \( \frac{1}{3}x^3 \). Easy, right? This powerful rule works every time, making integration approachable and much less intimidating.
In our exercise, the power rule transformed the integrand \( 7x^2 \) into \( \frac{7}{3}x^3 \), setting the stage for finding the definite integral between specific limits.
Indefinite Integral
Let's explore indefinite integrals, the mysterious relatives of definite integrals. Where definite integrals give us an actual number as a result, indefinite integrals are more about the process. They provide us with a function that represents all possible antiderivatives of the original function.
Think of indefinite integrals as not having boundaries; they roam free with a \( + C \), which represents an unknown constant. In technical terms, if you have a function \( f(x) \), the indefinite integral is symbolized as \( \int f(x) dx \), and it results in a new function \( F(x) + C \).
This \( F(x) \) is the antiderivative, or the original function before it was differentiated. In the context of our example, \( \frac{7}{3}x^3 + C \) is the antiderivative of \( 7x^2 \), which came about from applying the power rule in the integration process.
Think of indefinite integrals as not having boundaries; they roam free with a \( + C \), which represents an unknown constant. In technical terms, if you have a function \( f(x) \), the indefinite integral is symbolized as \( \int f(x) dx \), and it results in a new function \( F(x) + C \).
This \( F(x) \) is the antiderivative, or the original function before it was differentiated. In the context of our example, \( \frac{7}{3}x^3 + C \) is the antiderivative of \( 7x^2 \), which came about from applying the power rule in the integration process.
Integration Limits
Moving to integration limits, these are the markers that turn an indefinite integral into a definite one. Integration limits define the interval over which we’re integrating. They’re like the beginning and the end of a race—the ‘start’ and ‘finish’ lines for the integration process.
When you see \( \int_{a}^{b} f(x) dx \), the \( a \) and the \( b \) are the lower and upper limits, respectively. For definite integrals, we calculate the value of the antiderivative function at both limits. First, we plug in the upper limit and then subtract the value obtained with the lower limit plugged in.
In our textbook example, the limits were 1 and 3. We evaluated the antiderivative, \( \frac{7}{3}x^3 \), at these points to find our finite area under the curve, representing the definite integral between those specific points on the graph of the function.
When you see \( \int_{a}^{b} f(x) dx \), the \( a \) and the \( b \) are the lower and upper limits, respectively. For definite integrals, we calculate the value of the antiderivative function at both limits. First, we plug in the upper limit and then subtract the value obtained with the lower limit plugged in.
In our textbook example, the limits were 1 and 3. We evaluated the antiderivative, \( \frac{7}{3}x^3 \), at these points to find our finite area under the curve, representing the definite integral between those specific points on the graph of the function.
Significant Digits
Now let's dive into the world of precision with significant digits, also known as significant figures. These are not just numbers; they are a testament to the precision of a value. Significant digits convey how accurate a number is, based on the number of meaningful digits it includes.
In the realm of mathematics and science, we round off our answers to a certain number of significant figures to reflect the level of precision of the measurements or calculations we've made. If you're told to round an answer to three significant digits, you're aiming for the three most meaningful digits in the answer, starting from the first non-zero digit.
In the outcome of our integration exercise, the initial answer was a long decimal, but when constrained to three significant digits, it became a neat and precise 60.7. This provides a balance between accuracy and practicality, ensuring that our final answer is as faithful to the true value as our instructions allow.
In the realm of mathematics and science, we round off our answers to a certain number of significant figures to reflect the level of precision of the measurements or calculations we've made. If you're told to round an answer to three significant digits, you're aiming for the three most meaningful digits in the answer, starting from the first non-zero digit.
In the outcome of our integration exercise, the initial answer was a long decimal, but when constrained to three significant digits, it became a neat and precise 60.7. This provides a balance between accuracy and practicality, ensuring that our final answer is as faithful to the true value as our instructions allow.
