This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the integral and its limits. The given integral is: _0^1 _0^2 _0^sqrt(4-x^2) (x^2 + y^2) \, dy \, dx \, dz Step 2: Analyze the region of integration in the xy-plane. The limits for y are 0 y sqrt(4-x^2), which implies y^2 4-x^2 or x^2+y^2 4 (since y 0). The limits for x are 0 x 2. Combined, these limits describe the quarter-circle of radius 2 in the first quadrant of the xy-plane. Step 3: Convert to cylindrical coordinates. For the xy-plane, we use cylindrical coordinates: x = r y = r x^2 + y^2 = r^2 The differential dy \, dx becomes r \, dr \, d. The region 0 x 2, 0 y sqrt(4-x^2) in the xy-plane corresponds to: 0 r 2 (radius from origin to the circle x^2+y^2=4) 0 ()/(2) (first quadrant) The limits for z remain 0 z 1. Step 4: Rewrite the integral in cylindrical coordinates. Substitute the cylindrical coordinates into the integral: _0^1 _0^/2 _0^2 (r^2) \, r \, dr \, d \, dz _0^1 _0^/2 _0^2 r^3 \, dr \, d \, dz Step 5: Evaluate the innermost integral with respect to r. _0^2 r^3 \, dr = [ (r^4)/(4) ]_0^2 = (2^4)/(4) - (0^4)/(4) = (16)/(4) - 0 = 4 Step 6: Evaluate the middle integral with respect to . Now the integral becomes: _0^1 _0^/2 4 \, d \, dz _0^/2 4 \, d = [ 4 ]_0^/2 = 4 (()/(2)) - 4(0) = 2 Step 7: Evaluate the outermost integral with respect to z. Finally, the integral becomes: _0^1 2 \, dz = [ 2 z ]_0^1 = 2 (1) - 2 (0) = 2 The final answer is 2.