Evaluate the iterated integral $\int_0^4 \int_0^{e^v} \sqrt{1 + e^v} dw dv.$
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Step 1: First, we evaluate the inner integral with respect to $w$: $$ \int_0^{e^v} \sqrt{1 + e^v} \, dw = \sqrt{1 + e^v} \int_0^{e^v} dw = \sqrt{1 + e^v} [w]_0^{e^v} = \sqrt{1 + e^v} (e^v - 0) = e^v \sqrt{1 + e^v} $$ Show more…
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