Question

Evaluate the iterated integral ∫⁶₀∫ᵉ₀ √1+eᵛ dw dv

a) √e - 1
b) √e + 1
c) e - 1
d) e + 1

Answer 1

The iterated integral is evaluated by first performing the inner integral with respect to w, resulting in e(sqrt(1+e^v)) and then integrating with respect to v from 0 to 6, leading to the final answer e - 1.

Step-by-step explanation:

To evaluate the iterated integral ∫⁶₀∫ᴇ₀ √1+eᵗ dw dv, we integrate with respect to w first and then with respect to v. Since there are no w terms in the integrand, the integral ∫ᴇ₀ √1+eᵗ dw is simply w√1+eᵗ evaluated from 0 to e, which simplifies to e √1+eᵗ. The next step is to integrate this result with respect to v from 0 to 6.

The integral then becomes ∫⁶₀ e √1+eᵗ dv, which is a straightforward integral to evaluate. The result is e times the integral of √1+eᵗ, which can be found using a standard antiderivative for the square root of a sum of squares.

Once we integrate and evaluate from 0 to 6, the correct answer will be e - 1, which corresponds to answer option (c).