MATH 104 First Midterm Exam - Fall (d) A solid has as its base the region in the xy-plane the region between the curve y = 1 x2

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1 MATH 14 First Midterm Exam - Fall Find the area between the graphs of y = x 2 + x + 5 and y = 2x 2 x. 1. Find the area between the graphs of y = x 2 + 4x + 6 and y = 2x 2 x. 1. Find the area between the graphs of y = x 2 + 2x + 4 and y = 2x 2 x. 1. Find the area between the graphs of y = x 2 + x + and y = 2x 2 x (a) 24 (b) 6 (c) 4 6 (d) 5 6 (e) 2 (f) 81 2 y = 1 x2 4 y = 1 x4 16 y = 1 x2 9 y = 1 x4 81 (a) 8π (b) 2π (c) 2π 5 (d) 16π 45 (e) 8π 45 (f) 4π

2 . Let V (a) be the volume of the solid obtained by rotating the area between the graph x6 What is lim a. Let V (a) be the volume of the solid obtained by rotating the area between the graph x What is lim a. Let V (a) be the volume of the solid obtained by rotating the area between the graph x4 What is lim a. Let V (a) be the volume of the solid obtained by rotating the area between the graph x5 What is lim a (a) 4π (b) 2π (c) π 2 (d) π (e) 2π (f) π 2 4. Find the length of the part of the curve y = 1 4 x + 1 x 4. Find the length of the part of the curve y = 1 2 x + 1 6x 4. Find the length of the part of the curve y = 1 6 x + 1 2x 4. Find the length of the part of the curve y = 1 x + 1 x (a) 17 (b) 1 (c) 19 (d) 2 (e) 29 (f) 4

3 π/6 π/4 π/8 π/ cos(x) 1 + sin 2 (x) dx cos(2x) 1 + sin 2 (2x) dx cos(4x) 1 + sin 2 (4x) dx cos(6x) 1 + sin 2 (6x) dx (a) π 2 (b) π 4 (c) π 8 (d) π (e) π 16 (f) π An object moves in such a way that its acceleration is 2e t/2 meters per second 2. If 6. An object moves in such a way that its acceleration is 2e t/4 meters per second 2. If 6. An object moves in such a way that its acceleration is 6e t/2 meters per second 2. If 6. An object moves in such a way that its acceleration is 6e t/ meters per second 2. If (a) 9 m/s (b) 1 m/s (c) 14 m/s (d) 16 m/s (e) 17 m/s (f) 2 m/s

4 7. Find the surface area obtained by rotating the part of the curve y = x for x 2 around the y-axis. 7. Find the surface area obtained by rotating the part of the curve y = x 2 + for x around the y-axis. 7. Find the surface area obtained by rotating the part of the curve y = x for x 4 around the y-axis. 7. Find the surface area obtained by rotating the part of the curve y = x for x 1 around the y-axis. (a) π 6 (5 5 1) (b) π 6 ( ) (c) π 6 (7 7 1) (d) π 6 ( ) (e) π 6 ( ) (f) π 6 ( ) 8. Find the volume obtained by rotating the region between the curve y = 1 4x 2 8. Find the volume obtained by rotating the region between the curve y = 1 9x 2 8. Find the volume obtained by rotating the region between the curve y = 1 16x 2 8. Find the volume obtained by rotating the region between the curve y = 1 6x 2 (a) 2π 9 (b) π 4 (c) π (d) 4π 9 (e) 2π (f) π 2

5 9. The region inside the triangle with vertices (1, ), (1, 1) and (, ) is rotated around the line x = 1.What is the volume of the resulting solid? 9. The region inside the triangle with vertices (1, ), (1, 1) and (, ) is rotated around the line x = 2. What is the volume of the resulting solid? 9. The region inside the triangle with vertices (1, ), (1, 1) and (, ) is rotated around the line x =. What is the volume of the resulting solid? 9. The region inside the triangle with vertices (1, ), (1, 1) and (, ) is rotated around the line x = What is the volume of the resulting solid? (a) 2π (b) 5π (c) 8π (d) 11π (e) 14π (f) 17π 1. The region between the x-axis, the y-axis, and the line x = 8 8x is revolved around 1. The region between the x-axis, the y-axis, and the line y = 2 2x is revolved around 1. The region between the x-axis, the y-axis, and the line x = 4 4x is revolved around 1. The region between the x-axis, the y-axis, and the line x = 6 6x is revolved around (Hint: Use disks. What is the center of mass of a disk? What fraction of the total mass is in each disk? It will save an integral to know that the volume of a cone is 1 πr2 h, where r is the radius of the base of the cone and h is its height.) (a) 8 (b) 1 2 (c) 2 (d) 1 (e) 2 (f) 4