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MATH 101 HOMEWORK 8 Due on Wednesday, November 12 Covers sections 7.1–7.3. For full credit, show all work. 1. (6 marks) A bowl is shaped like a hemisphere with diameter 30 cm. A ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl. (Note: you may have to consider the cases h ≤ 10 and h > 10 separately.) 2. (6 marks) A notch is cut to the centre of a cylindrical log having radius 20 cm. One plane face of the notch is perpendicular to the axis of the log. The other one intersects it along a diameter of the log at an angle of 30 deg. (See Figure 7.20 in the textbook.) What volume of wood was removed from the log by cutting the notch? 3. Set up and evaluate the integrals for: (a) (4 marks) the length of the curve y = ex + e−x between x − 0 and x = a, (b) (4 marks) the area of the surface obtained by rotating the curve y = e2x , 0 ≤ x ≤ a, around the x-axis. 1