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In this video I determine the volume of a region whose distance is at most 1 from a box. To understand how to solve this, I solve the 2D case first and solve the area around a rectangle. In the 2D case, the region can be divided into 5 rectangles and 4 quarter circles. In the 3D case, the region can be divided into 7 rectangles, 12 quarter cylinders, and 8 one-eighth spheres. Summing all these up we obtain the volume in the 3D case and area in the 2D case.
Timestamps:
Problem 2: Volume around a box: 0:00
Solution: Solve the 2D case first: 0:42
Area around a square is LW + 2L + 2W + pi
Solving the 3D case: 3:54
Volume around a box is LWH + 2(LW + LH + WH) + pi(L + W + H) + 4/3 pi: 14:51
Problems Plus 2: Volume Around a Box (and Area Around a Rectangle)