Problem 174
Question
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{2} x^{9} d x $$
Step-by-Step Solution
Verified Answer
The value of the definite integral is 102.3.
1Step 1: Identify the Integral Expression
The integral expression given is \( \int_{1}^{2} x^{9} \, dx \). This is a definite integral from 1 to 2, involving the function \( x^9 \).
2Step 2: Apply the Power Rule for Integration
According to the power rule, to integrate a function of the form \( x^n \), we use the formula \( \frac{x^{n+1}}{n+1} \). Applying this to \( x^9 \), we get \( \frac{x^{10}}{10} \).
3Step 3: Evaluate the Indefinite Integral at the Upper Limit
Substitute the upper limit, 2, into the integrated function: \( \frac{2^{10}}{10} = \frac{1024}{10} = 102.4 \).
4Step 4: Evaluate the Indefinite Integral at the Lower Limit
Substitute the lower limit, 1, into the integrated function: \( \frac{1^{10}}{10} = \frac{1}{10} = 0.1 \).
5Step 5: Compute the Definite Integral
Subtract the value obtained at the lower limit from the value obtained at the upper limit: \( 102.4 - 0.1 = 102.3 \). This is the value of the definite integral.
Key Concepts
Definite IntegralPower Rule for IntegrationUpper and Lower LimitsEvaluate Integral
Definite Integral
A definite integral is a fundamental concept in calculus that represents the accumulated area under a curve of a function over a specific interval. In our example, the definite integral is denoted by \( \int_{1}^{2} x^{9} \, dx \), where the numbers 1 and 2 are known as the limits of integration.
The definite integral can be visualized as calculating the total area under the curve \( x^9 \), from \( x = 1 \) to \( x = 2 \). In simpler terms, it gives us a number that represents this accumulated area, which helps in various applications like determining distance, probability, and other measurable quantities.
- The lower limit is the starting point of the interval, referred to as \( x = 1 \) in this case.
- The upper limit is the ending point of the interval, here \( x = 2 \).
The definite integral can be visualized as calculating the total area under the curve \( x^9 \), from \( x = 1 \) to \( x = 2 \). In simpler terms, it gives us a number that represents this accumulated area, which helps in various applications like determining distance, probability, and other measurable quantities.
Power Rule for Integration
The power rule for integration is a basic technique used to integrate polynomial functions. It is similar to the power rule for differentiation but applied in the reverse process.
To integrate a function of the form \( x^n \), where \( n eq -1 \), use the following formula:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In our exercise, this rule is applied to the function \( x^9 \). Thus, the integral becomes:
The constant \( C \) is not required here since we are dealing with a definite integral.
To integrate a function of the form \( x^n \), where \( n eq -1 \), use the following formula:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In our exercise, this rule is applied to the function \( x^9 \). Thus, the integral becomes:
- \( \int x^9 \, dx = \frac{x^{10}}{10} + C \)
The constant \( C \) is not required here since we are dealing with a definite integral.
Upper and Lower Limits
The concept of upper and lower limits is crucial in calculating definite integrals. These bounds dictate the span of the function's domain where the area is to be evaluated. In the problem \( \int_{1}^{2} x^{9} \, dx \):
Using these limits, we substitute them into the integrated expression and evaluate it at these points:
- \( x = 1 \) is the lower limit, marking the beginning of the interval.
- \( x = 2 \) is the upper limit, indicating the end of the interval.
Using these limits, we substitute them into the integrated expression and evaluate it at these points:
- Substitute 2 into \( \frac{x^{10}}{10} \) to get \( \frac{2^{10}}{10} = 102.4 \).
- Substitute 1 into \( \frac{x^{10}}{10} \) to get \( \frac{1}{10} = 0.1 \).
Evaluate Integral
Once we have plugged in the upper and lower limits into our integrated expression, we need to compute the difference to find the value of the definite integral. This process effectively gives us the net area between the function curve and the x-axis over the specified interval.
For the integral \( \int_{1}^{2} x^{9} \, dx \):
To find the definite integral, subtract the lower limit evaluation from the upper limit evaluation:
\( 102.4 - 0.1 = 102.3 \).
This result, \( 102.3 \), represents the area below the curve \( x^9 \) from \( x = 1 \) to \( x = 2 \).
For the integral \( \int_{1}^{2} x^{9} \, dx \):
- Value at the upper limit 2: \( 102.4 \)
- Value at the lower limit 1: \( 0.1 \)
To find the definite integral, subtract the lower limit evaluation from the upper limit evaluation:
\( 102.4 - 0.1 = 102.3 \).
This result, \( 102.3 \), represents the area below the curve \( x^9 \) from \( x = 1 \) to \( x = 2 \).
Other exercises in this chapter
Problem 172
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