Let's solve each problem step-by-step: --- ### **1. Find the area under \( f(x) = 9e^x \) on \([0, 3]\)** The area under the curve is given by the definite integral: \[ \int_{0}^{3} 9e^x \, dx \] **Step 1: Integrate the function** The antiderivative of \( 9e^x \) is \( 9e^x \). **Step 2: Evaluate from 0 to 3** \[ \left[ 9e^x \right]_{0}^{3} = 9e^{3} - 9e^{0} = 9e^3 - 9 \cdot 1 = 9e^3 - 9 \] **Final Answer:** \[ \boxed{9e^3 - 9} \text{ square units} \] --- ### **2. Evaluate \( \int_{0}^{1} 4x^3 \, dx \)** **Step 1: Integrate the function** The antiderivative of \( 4x^3 \) is \( x^4 \). **Step 2: Evaluate from 0 to 1** \[ \left[ x^4 \right]_{0}^{1} = 1^4 - 0^4 = 1 - 0 = 1 \] **Final Answer:** \[ \boxed{1} \] --- ### **3. Evaluate \( \int_{1}^{8} \frac{4}{x} \, dx \)** **Step 1: Integrate the function** The antiderivative of \( \frac{4}{x} \) is \( 4\ln|x| \). **Step 2: Evaluate from 1 to 8** \[ \left[ 4\ln|x| \right]_{1}^{8} = 4\ln(8) - 4\ln(1) = 4\ln(8) - 0 = 4\ln(8) \] **Final Answer:** \[ \boxed{4\ln(8)} \] --- #### **Summary of Answers** 1. \( 9e^3 - 9 \) square units 2. \( 1 \) 3. \( 4\ln(8) \) Let me know if you need further explanation!