Integral of 2x dx from 10 to 13
Understand the Problem
The question is asking for the definite integral of the function 2x with respect to x, calculated from the limits 10 to 13. This involves finding the antiderivative of 2x and then evaluating it at the specified limits.
Answer
The definite integral is \(69\).
Answer for screen readers
The value of the definite integral is (69).
Steps to Solve
- Find the Antiderivative
To solve for the definite integral of (2x), we first determine the antiderivative. The antiderivative of (2x) is:
$$ F(x) = x^2 + C $$
where (C) is the constant of integration.
- Evaluate at the Limits
Next, we need to evaluate (F(x)) at the upper limit (13) and lower limit (10) and subtract the two results:
$$ F(13) - F(10) $$
- Calculate (F(13))
Now, calculate (F(13)):
$$ F(13) = 13^2 = 169 $$
- Calculate (F(10))
Next, calculate (F(10)):
$$ F(10) = 10^2 = 100 $$
- Subtract the Results
Now, subtract (F(10)) from (F(13)):
$$ F(13) - F(10) = 169 - 100 = 69 $$
The value of the definite integral is (69).
More Information
The definite integral represents the net area under the curve (y = 2x) from (x = 10) to (x = 13). The result of (69) indicates that the area above the x-axis is greater than that below it in this interval.
Tips
- Forgetting to evaluate the antiderivative at both limits can lead to incomplete calculations.
- Miscalculating the squares of the limits can result in incorrect values.
- Not subtracting the lower limit's evaluation from the upper limit's correctly.
