This question is asking to compute the exact area under the curve $f(x) = x$ over the interval $[4, 10]$ using a definite integral. Let’s work through it step by step:

1. Set up the definite integral:

The definite integral of $f(x) = x$ from $4$ to $10$ is: $\int_{4}^{10} x \, dx$

2. Integrate $f(x) = x$:

The antiderivative of $x$ is: $\frac{x^2}{2}$

3. Evaluate the definite integral:

Now apply the limits of integration: $\int_{4}^{10} x \, dx = \left[ \frac{x^2}{2} \right]_4^{10}$ This means: $\frac{10^2}{2} - \frac{4^2}{2}$

4. Simplify:

$\frac{10^2}{2} = \frac{100}{2} = 50, \quad \frac{4^2}{2} = \frac{16}{2} = 8$ So: $50 - 8 = 42$

5. Answer:

The exact area under the curve is 42.

Would you like additional details about definite integrals or further explanations?

Here are 5 follow-up questions to expand understanding:

1. What is the geometric interpretation of a definite integral?
2. How would the solution change if the interval was $[0, 10]$?
3. Can you generalize the formula for the integral of $f(x) = mx + b$?
4. What if $f(x)$ was a quadratic function instead of linear?
5. How does the Fundamental Theorem of Calculus relate to solving this problem?

Tip: Always double-check the limits of integration when solving definite integrals—they define the portion of the curve you’re calculating!