Evaluate the integral, where E is enclosed by the paraboloid z = 7+ x2 + y?, the cylinder x? + y? = 5, and the xy-plane. Use cylindrical coordinates. e dv Step 1 In cylindrical coordinates, the paraboloid z = 7 + x? + y? has the equation z = 9+ y2 7+ 2 and the cylinder x + y = 5 has the equation r=V5 Step 2 Therefore, the region E enclosed by the paraboloid, the cylinder, and the xy-plane is described by E = {r, 0, 2)| 0 s zs 7+r OsrsV5 Osos 27 7+ 2 Step 3 To evaluate e dv= er dz dr de, we first calculate the innermost integral. er dz =

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