Question

The value of the integral \[ \iiint\limits_{0}^{a, b, c} e^{x+y+z} \, dz \, dy \, dx \] is:

• A. \( e^{a+b+c} \)
• B. \( e^a + e^b + e^c \)
• C. \( (e^a -1)(e^b -1)(e^c -1) \)
• D. \( e^{abc} \)

Hint:

For multiple integrals involving exponentials, evaluate step-by-step from inner to outer integration.

Answer 1

The Correct Option is C

Step 1: Compute inner integral. \[ \int_0^c e^{x+y+z} dz = e^{x+y} \int_0^c e^z dz = e^{x+y} [e^c -1]. \]
Step 2: Compute second integral. \[ \int_0^b e^{x+y} (e^c -1) dy = (e^c -1) e^x \int_0^b e^y dy = (e^c -1) e^x [e^b -1]. \]
Step 3: Compute final integral. \[ \int_0^a (e^c -1)(e^b -1) e^x dx = (e^c -1)(e^b -1) [e^a -1]. \] Thus, the integral evaluates to: \[ (e^a -1)(e^b -1)(e^c -1). \]
Step 4: Selecting the correct option. Since \( (e^a -1)(e^b -1)(e^c -1) \) matches, the correct answer is (C).