21b_ps6 - 3 , y = 1 2 t 2 for ! t ! 1 6) By parameterizing...

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Co-21B Problem Solving #6T 1) SET UP integral(s) that could be used to calculate the volume of the solid obtained by revolving the region bounded by y = x 3 , y = 2 ! x , and y = 0 about the x – axis using the: a) Disc Method b) Shell Method 2) SET UP integrals to evaluate the volume obtained by revolving the region bounded by y = 4 x ! x 2 and y = 0 about: a) the y –axis b) x = 10 c) y = -2 3) If f ( x ) is continuous, and f ( x ) dx = 12 0 4 ! , find: a) f (2 x ) dx 0 2 ! b) f ( x ) dx x 0 16 ! 4) A water tank in the shape of a hemisphere with a radius of 6 feet is filled with water to a depth of 3 feet. Find the volume of water in the tank. 5) Find the length of each curve: a) y = x 5/4 from (0, 0) to (1, 1) b) y = x 2 ! ln x 8 , for 1 ! x ! 2 c) x = y 3 6 + 1 2 y for 2 ! y ! 3 c) x = 1 3 t
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Unformatted text preview: 3 , y = 1 2 t 2 for ! t ! 1 6) By parameterizing the x and y coordinates of a circle of radius r , prove that the length around a circle is 2 ! r . 7) Integration practice: a) ln x x (1 + ln x ) dx ! b) x 2 x + 1 dx ! c) x + tan ! 1 x 1 + x 2 dx " d) x + 1 (3 x + 2) 10 dx ! 8) The base of a solid is an isosceles triangle whose base is 4 feet and height is 5 feet. Cross-sections perpendicular to the altitude are semi-circles. Find the volume using: a) an integral b) common sense 9) Find the volume of the solid formed when the area between y = 1 1 + x 2 , y = x x + 1 , x = , and x = 1 is revolved about the x axis....
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