Math Problem Statement
Solution
Let’s solve the given iterated integral:
Problem:
Step 1: Separate the Exponential Function
The expression can be written as: This allows the integral to separate as a product of two simpler integrals:
= \int_{0}^{\ln 3} e^{4x} \, dx \cdot \int_{1}^{\ln 4} e^{2y} \, dy$$ --- ### Step 2: Evaluate Each Integral #### (a) $$\int_{1}^{\ln 4} e^{2y} \, dy$$ The integral of $$e^{2y}$$ with respect to $$y$$ is: $$\int e^{2y} \, dy = \frac{e^{2y}}{2}$$ Applying the limits $$y = 1$$ to $$y = \ln 4$$: $$\left[ \frac{e^{2y}}{2} \right]_{1}^{\ln 4} = \frac{e^{2 \ln 4}}{2} - \frac{e^{2 \cdot 1}}{2}$$ Simplify: $$e^{2 \ln 4} = (e^{\ln 4})^2 = 4^2 = 16, \quad e^{2 \cdot 1} = e^2$$ $$\int_{1}^{\ln 4} e^{2y} \, dy = \frac{16}{2} - \frac{e^2}{2} = 8 - \frac{e^2}{2}$$ #### (b) $$\int_{0}^{\ln 3} e^{4x} \, dx$$ The integral of $$e^{4x}$$ with respect to $$x$$ is: $$\int e^{4x} \, dx = \frac{e^{4x}}{4}$$ Applying the limits $$x = 0$$ to $$x = \ln 3$$: $$\left[ \frac{e^{4x}}{4} \right]_{0}^{\ln 3} = \frac{e^{4 \ln 3}}{4} - \frac{e^{4 \cdot 0}}{4}$$ Simplify: $$e^{4 \ln 3} = (e^{\ln 3})^4 = 3^4 = 81, \quad e^{4 \cdot 0} = 1$$ $$\int_{0}^{\ln 3} e^{4x} \, dx = \frac{81}{4} - \frac{1}{4} = \frac{80}{4} = 20$$ --- ### Step 3: Combine the Results Now, multiply the results of the two integrals: $$\int_{0}^{\ln 3} \int_{1}^{\ln 4} e^{4x + 2y} \, dy \, dx = \left( 20 \right) \cdot \left( 8 - \frac{e^2}{2} \right)$$ Distribute $$20$$: $$20 \cdot \left( 8 - \frac{e^2}{2} \right) = 20 \cdot 8 - 20 \cdot \frac{e^2}{2} = 160 - 10e^2$$ --- ### Final Answer: $$\boxed{\text{a. } 10(16 - e^2)}$$ --- Would you like a more detailed explanation of any step? Here are 5 related questions to deepen your understanding: 1. How do you separate exponents when combining variables in $$e^{a+b}$$? 2. Why is the integral $$\int e^{ax} \, dx = \frac{e^{ax}}{a}$$? 3. What is the property of logarithms used to simplify $$e^{\ln a} = a$$? 4. Can you apply a similar technique to solve triple integrals? 5. How does the choice of integration order affect the calculation? **Tip:** Always check if the integral can be separated when the exponent is a sum of independent variables. It simplifies the problem significantly!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Double Integrals
Exponential Functions
Integration by Substitution
Formulas
\( e^{a+b} = e^a \cdot e^b \)
\( \int e^{ax} \, dx = \frac{e^{ax}}{a} \)
Theorems
Property of Exponents
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate Calculus
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