Integrate 2x dx from 10 to 13.
Understand the Problem
The question is asking to calculate the definite integral of the function 2x with respect to x, from the limits 10 to 13. This involves finding the antiderivative of 2x and then evaluating it at the upper and lower limits.
Answer
$69$
Answer for screen readers
The final answer is $69$.
Steps to Solve
- Finding the Antiderivative
To solve the definite integral $\int_{10}^{13} 2x , dx$, we first need to find the antiderivative of the function $2x$.
The antiderivative of $2x$ is given by:
$$ \int 2x , dx = x^2 + C $$
where $C$ is the constant of integration.
- Evaluating the Antiderivative at the Limits
Now, we will evaluate the antiderivative at the upper limit (13) and the lower limit (10).
First, we evaluate at the upper limit:
$$ F(13) = (13)^2 = 169 $$
Next, we evaluate at the lower limit:
$$ F(10) = (10)^2 = 100 $$
- Calculating the Definite Integral
Finally, we calculate the definite integral by subtracting the value at the lower limit from the value at the upper limit:
$$ \int_{10}^{13} 2x , dx = F(13) - F(10) = 169 - 100 = 69 $$
The final answer is $69$.
More Information
The definite integral calculates the area under the curve of the function $2x$ from $x=10$ to $x=13$. The value $69$ represents the total area between the curve and the x-axis across that interval.
Tips
- Forgetting to apply the limits correctly or miscalculating the squares when evaluating the antiderivative at the limits. Always double-check arithmetic after substitution.
