Here is a transcription of the question: --- **9. (15 points) Consider the solid of revolution given by revolving \(y = x^2 + \ln(x)\) from \(x = e\) to \(x = 2e\) about the x-axis. Write down a definite integral for the volume of the solid of revolution. Provide your answer by writing down a definite integral in the box for Question 9 on the answer sheet. Do not justify your work or evaluate the integral.** --- ### **Setting Up the Integral** For a solid of revolution about the x-axis, the volume \( V \) is given by: \[ V = \pi \int_{a}^{b} [f(x)]^2 dx \] Here, \( f(x) = x^2 + \ln(x) \), \( a = e \), \( b = 2e \). ### **The Definite Integral** \[ \boxed{ V = \pi \int_{e}^{2e} \left[x^2 + \ln(x)\right]^2 dx } \] That's the integral you should write as your answer.

Certainly! Here are the three theoretical steps with one-line explanations, followed by the final integral answer: **Step 1:** Identify the function \(f(x) = x^2 + \ln(x)\) and the limits \(x = e\) to \(x = 2e\). **Step 2:** Use the volume of revolution formula about the x-axis: \(V = \pi \int_{a}^{b} [f(x)]^2 dx\). **Step 3:** Set up the definite integral as \(V = \pi \int_{e}^{2e} (x^2 + \ln(x))^2 dx\). **Final Answer:** \[ \boxed{ V = \pi \int_{e}^{2e} \left[x^2 + \ln(x)\right]^2 dx } \]