Evaluate the following integral from 1 to 2 of (5 - 0) dθ.
Understand the Problem
The question is asking to evaluate a definite integral from 1 to 2 of the function 5 - 0 with respect to the variable dθ. This requires using integral calculus to find the area under the curve represented by the function over the specified range.
Answer
The result of the integral is $$ 5 $$
Answer for screen readers
The value of the definite integral is
$$ 5 $$
Steps to Solve
- Identify the Integral Expression The integral to evaluate is
$$ \int_{1}^{2} (5 - 0) , d\theta $$
This simplifies to
$$ \int_{1}^{2} 5 , d\theta $$
- Integrate the Expression Integrating the constant $5$ with respect to $\theta$ gives:
$$ 5\theta + C $$
- Evaluate the Integral with Limits Now, evaluate the definite integral from $\theta = 1$ to $\theta = 2$:
$$ \left[ 5\theta \right]_{1}^{2} = 5(2) - 5(1) $$
- Calculate the Final Values Continuing from the last step, calculate the final values:
$$ = 10 - 5 = 5 $$
The value of the definite integral is
$$ 5 $$
More Information
This integral represents the area under the curve (which is a constant in this case) between the specified limits. The result signifies the total area between the function and the horizontal axis from $\theta = 1$ to $\theta = 2$.
Tips
- Misinterpreting the expression: Some might confuse $5 - 0$ as a complex function instead of recognizing it as a constant.
- Forgetting to apply the limits correctly: It’s vital to substitute both limits in the evaluated function.
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