advertisement

MATH 101 V2A February 4th – Practice problems Hints and Solutions Method of “washers”: 1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. (a) y 2 = x, x = 2y; about the y-axis. Solution: We want to calculate the volume of the solid obtained by rotating the shaded region around the y-axis. y 2y = x √ y= x x 4 From this we can see that a washer will have inner radius r = y 2 and outer radius R = 2y. This means that the volume of the solid is Z 2 Z 2 4 3 1 5 2 64 2 2 2 2 4 V =π (2y) − (y ) dy = π (4y − y )dy = π y − y = π. 3 5 15 0 0 0 (b) y = x, y = √ x; about x = 2. Solution: We want to calculate the volume of the solid obtained by rotating the shaded region about the line x = 2 y = x √ x = y y x 1 2 From this we can see that a washer will have inner radius r = 2 − y and outer radius R = 2 − y 2 . This means that the volume of the solid is Z 1 Z 1 1 5 1 8 V =π (2 − y 2 )2 − (2 − y)2 dy = π (−5y 2 +y 4 +4y)dy = π − y 3 + y 5 + 2y 2 = π. 3 5 15 0 0 0 2. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. (a) y = 0, y = sin(x) for 0 ≤ x ≤ π; about y = −2. Z Hint: Answer: V = π (2 + sin(x))2 − 22 dx. 0 (b) x2 − y 2 = 1, x = 3; about x = −2. √ Z Hint: Use the symmetry of the solid to simplify the integral. Answer: V = 2π 8 52 − (2 + p y 2 − 1)2 dy. 0 Method of cylindrical shells: 1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. (a) y = x2 , y = 0, x = 1, x = 2; about x = 1. Solution: We want to calculate the volume of the solid obtained by rotating the shaded region about the line x = 1. 2 y y = x2 x 1 From this we can see that a cylindrical shell will have radius (x − 1) and height x2 . Therefore, Z 2 2 1 √ 2 3 x −x (x − 1) · x dx = 2π V = 2π (b) y = Z 2 dx = 2π 1 1 4 1 3 2 17 x − x = π. 4 3 1 6 x − 1, y = 0, x = 5; about y = 3. Solution: We want to calculate the volume of the solid obtained by rotating the shaded region about the line y = 3. y 3 y= √ x−1 x 5 From this we can see that a cylindrical shell will have radius (3 − y) and height 5 − (y 2 + 1). Therefore Z 2 Z 2 1 4 2 2 2 3 3 2 V = 2π (3−y)(5−(y +1))dy = 2π 12 − 3y − 4y + y dy = 2π 12y − y − 2y + y = 24π. 4 0 0 0 2. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. 3 (a) y = log(x), y = 0, x = 2; about the y-axis. Z log(2) y(2 − ey )dy. Hint: Answer: V = 2π 0 (b) y = 1 1+x2 , y = 0, x = 0, x = 2; about x = 2. Z 2 (2 − x) · Hint: Answer: V = 2π 0 1 dx. 1 + x2 4