{
  "problem": "Evaluate the iterated integral. $ \\int_{-1}^1 \\left( \\int_{-2}^2 e^x + e^y \\, dy \\right) dx =$ Choose 1 answer: Choose 1 answer: (Choice A) A $2e^2 + 4e - 4e^{-1} - 2e^{-2}$ (Choice B) B $4e^2 + 4e^{-2}$ (Choice C) C $2e^2 - 2e^{-2}$ (Choice D) D $e^2 - 4e - 4e^{-1} + e^{-2}$",
  "hints": [
    "Evaluate the inner integral: $\\begin{aligned} \\int_{-1}^1 \\left( \\int_{-2}^2 e^x + e^y \\, dy \\right) dx &= \\int_{-1}^1 \\left[ ye^x + e^y \\right]_{-2}^2 \\, dx \\\\ \\\\ &= \\int_{-1}^1 2e^x + e^2 - \\left( -2e^x + e^{-2} \\right) \\, dy \\\\ \\\\ &= \\int_{-1}^1 4e^x + e^2 - e^{-2} \\, dy \\end{aligned}$",
    "Evaluate the outer integral: $\\begin{aligned} \\int_{-1}^1 4e^x + e^2 - e^{-2} \\, dy &= 4e^x + x \\left( e^2 - e^{-2} \\right) \\bigg|_{-1}^1 \\\\ \\\\ &= 4e + e^2 - e^{-2} - \\left( 4e^{-1} - e^2 + e^{-2} \\right) \\\\ \\\\ &= 2e^2 + 4e - 4e^{-1} - 2e^{-2} \\end{aligned}$",
    "The answer: $ \\int_{-1}^1 \\left( \\int_{-2}^2 e^x + e^y \\, dy \\right) dx = 2e^2 + 4e - 4e^{-1} - 2e^{-2}$"
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/compute-iterated-integrals/16078041242.json'}