Math 2414 DHW 3

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Mathematics

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Apr 3, 2024

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Ashwin Indurti Math2414SP24 Assignment DHW 3 6.3 - 7.1 S24 due 02/05/2024 at 11:59pm CST Problem 1. (1 point) Find the volume of the solid obtained by rotating the region bounded by y = 1 x , x = 2 , x = 8 , and y = 0 about the y -axis. V = Answer(s) submitted: 37.699 (correct) Problem 2. (1 point) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves y = 8 x 2 , y = 0 , x = 1 , and x = 2 about the y -axis. V = Answer(s) submitted: 60pi (correct) Problem 3. (1 point) Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by y = e - 9 x 2 , y = 0 , x = 0 , and x = 8 about the y -axis. V = Answer(s) submitted: (pi(eˆ576-1))/(9eˆ576) (correct) Problem 4. (1 point) Find the volume of the solid obtained by rotating the region bounded by y = 4cos ( x 2 ) , y = 0, x = 0, and x = r π 3 about the y -axis. V = Answer(s) submitted: (4pisqrt3)/2 (correct) Problem 5. (1 point) Find the volume of the solid obtained by rotating the region bounded by the curves x = y , x = 0 , and y = 9 about the x -axis. V = Answer(s) submitted: (972pi)/5 (correct) Problem 6. (1 point) Find the volume of the solid obtained by rotating the region bounded by the curves x = 1 + y 2 , x = 0 , y = 3 , and y = 4 about the x -axis. V = Answer(s) submitted: (189pi)/2 (correct) Problem 7. (1 point) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves x = 4 y 2 - y 3 and x = 0 about the x -axis. V = Answer(s) submitted: (512pi)/5 (correct) Problem 8. (1 point) Use the shell method to compute the volume of the solid obtained by rotating the region enclosed by the functions y = x 2 , y = 2 - x 2 and to the right of x = 0 about the y -axis. V = Answer(s) submitted: pi (correct) Problem 9. (1 point) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by y = 4 ( x - 2 ) 2 and y = x 2 - 4 x + 7 about the y -axis. V = Answer(s) submitted: 16pi (correct) Problem 10. (1 point) Let V be the volume of the solid obtained by rotating about the y -axis the region bounded by y = 2 x and y = 2 x 2 . Find V either by the disk/washer method or by cylindrical shells. V = Answer(s) submitted: (3pi)/5 (correct) 1
Problem 11. (1 point) Evaluate the integral. Z x cos ( 2 x ) dx = Answer(s) submitted: (1/4)(2xsin(2x) + cos(2x) + c) (correct) Problem 12. (1 point) Evaluate the integral. Z xe 4 x dx = . Answer(s) submitted: 1/16(4x(eˆ(4x))-eˆ(4x))+c (correct) Problem 13. (1 point) Evaluate the integral. Z 4 1 - 3 t ln ( t ) dt = Answer(s) submitted: 28/3 - 32ln2 (correct) Problem 14. (1 point) Evaluate the integral: Z 1 0 - 4 y e 2 y dy = Answer(s) submitted: (3/eˆ2)-1 (correct) Problem 15. (1 point) Suppose that f ( 1 ) = 10 , f ( 4 ) = - 4 , f 0 ( 1 ) = - 9 , f 0 ( 4 ) = - 8 , and f 00 is continuous. Use this to evaluate the integral. Z 4 1 x f 00 ( x ) dx = Answer(s) submitted: -32+23 (correct) Problem 16. (1 point) Evaluate the integral: Z 5 t 3 e t dt = Answer(s) submitted: 5((tˆ3*eˆt)-3((tˆ2*eˆt)-2((eˆt*t)-eˆt)))+c (correct) Problem 17. (1 point) Evaluate the integral. Z 1 0 5 ( x 2 + 1 ) e - x dx = Answer(s) submitted: 5((-6/e)+3) (correct) Problem 18. (1 point) Evaluate the integral. Z π / 2 π / 4 3 x csc 2 ( x ) dx = Answer(s) submitted: -3((-pi/4)-((1/2)ln2)) (correct) Problem 19. (1 point) Evaluate the integral. Z - 2 x 2 sin ( π x ) dx = Answer(s) submitted: -2((-1/pi)xˆ2cos(pix)+ (2/pi)((1/pi)xsin(pix)+(1/piˆ2)cos( (correct) Problem 20. (1 point) Evaluate the integral. Z x sec 2 ( 7 x ) dx = Answer(s) submitted: 1/2xˆ2secˆ2(7x)-1/98(49xˆ2tanˆ2(7x)-14xtan(7x)-2ln|cos(7x) (correct) Problem 21. (1 point) Evaluate the integral. Z sin - 1 ( 5 x ) dx = Answer(s) submitted: xarcsin(5x)+1/5sqrt(1-25xˆ2)+c (correct) 2
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