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MATH 101 HOMEWORK 8 Due on Wednesday Nov. 17, 2004. For full credit, show all work. Calculators are not allowed, except as indicated below. 1. (5 marks) A notch is cut to the centre of a cylindrical log of 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 60 degrees. (See Figure 7.20 in the textbook.) What volume of wood was removed from the log by cutting the notch? 2. (5 marks) Use Simpson’s Rule with n = 10 to find the approximate length of the curve y = x3 between x = 0 and x = 1. (You may use a calculator to find the numerical answer, but you first have to write out the sum that you will be evaluating.) 3. (5 marks) Find the surface area of the torus obtained by rotating the circle (x−2)2 +y 2 = 1 around the y axis. 4. (5 marks) A circular cone standing upright on its base has base radius 10 cm and height 30 cm. It is made of material that compresses under pressure, so that its density at height x (measured from the base) is 2 + 10−4 (30 − h)2 g/cm3 . Find the mass of the cone. (You may use a calculator to find the numerical answer after you set up the integral.) This homework covers Sections 7.2–7.3 and part of 7.4. Additional recommended exercises: Section 7.2: 1–19; Section 7.3: 1–31; Section 7.4: 1–14. 1