Surface Area

Question 1

What is the formula for surface area?

Answer 1

2pi int f(x)sqrt(1 + f’(x)^2)

Question 3

y = x^3 from x = 0 to x = 1; rotate about x-axis.

Answer 3

S = 2pi * integral[0,1] x^3 * sqrt(1 + 9x^4) dx. Result: (pi/27)(10*sqrt(10) - 1).

Question 4

y = sqrt(x) from x = 1 to x = 4; rotate about x-axis.

Answer 4

S = 2pi * integral[1,4] sqrt(x) * sqrt(1 + 1/4x) dx = pi * integral sqrt(4x+1). Result: (pi/6)(17sqrt(17) - 5sqrt(5)).

Question 5

y = sin(x) from x = 0 to x = pi; rotate about x-axis.

Answer 5

S = 2pi * integral[0,pi] sin(x) * sqrt(1 + cos^2 x) dx. Result: 2pi[sqrt(2) + ln(1 + sqrt(2))].

Question 6

y = x^2 from x = 0 to x = sqrt(2); rotate about y-axis.

Answer 6

S = 2pi * integral[0,sqrt(2)] x * sqrt(1 + 4x^2) dx. Result: (13pi/6).

Question 7

y = e^x from x = 0 to x = 1; rotate about x-axis.

Answer 7

S = 2pi * integral[0,1] e^x * sqrt(1 + e^(2x)) dx. Result: pi[e*sqrt(1+e^2) + ln(e + sqrt(1+e^2)) - …].

Question 8

y = sqrt(4 - x^2) from x = -1 to x = 1; rotate about x-axis.

Answer 8

S = 2pi * integral[-1,1] sqrt(4-x^2) * (2/sqrt(4-x^2)) dx. Result: 8pi.

Question 9

y = x + 1 from x = 0 to x = 1; rotate about x-axis.

Answer 9

S = 2pi * integral[0,1] (x+1)sqrt(2) dx. Result: 3pisqrt(2).

Question 10

y = (1/3)x^3 from x = 0 to x = 3; rotate about x-axis.

Answer 10

S = 2pi * integral[0,3] (1/3)x^3 * sqrt(1 + x^4) dx. Result: (pi/9)(82*sqrt(82) - 1).

Question 11

x = sqrt(y) from y = 0 to y = 4; rotate about y-axis.

Answer 11

S = 2pi * integral[0,4] sqrt(y) * sqrt(1 + 1/4y) dy. Result: (pi/6)(17*sqrt(17) - 1).

Question 12

y = cosh(x) from x = 0 to x = 1; rotate about x-axis.

Answer 12

S = 2pi * integral[0,1] cosh^2(x) dx. Result: pi(1 + sinh(2)/2).

Question 13

y = (1/4)x^4 + (1/8x^2) from x = 1 to x = 2; rotate about x-axis.

Answer 13

S = 2pi * integral (y * ds). Result: (169pi/16).

Question 14

y = 2*sqrt(x) from x = 1 to x = 2; rotate about x-axis.

Answer 14

S = 2pi * integral[1,2] 2sqrt(x) * sqrt(1 + 1/x) dx. Result: 8pi/3(3sqrt(3) - 2*sqrt(2)).

Question 15

y = sqrt(x + 1) from x = 1 to x = 5; rotate about x-axis.

Answer 15

S = 2pi * integral sqrt(x+1)sqrt(1+1/(4(x+1))) dx. Result: (pi/6)(21sqrt(21) - 9*sqrt(9)).

Question 16

x = y^3/3 from y = 0 to y = 1; rotate about y-axis.

Answer 16

S = 2pi * integral[0,1] (y^3/3) * sqrt(1 + y^4) dy. Result: (pi/9)(2*sqrt(2) - 1).

Question 17

y = 1 - x^2 from x = 0 to x = 1; rotate about y-axis.

Answer 17

S = 2pi * integral[0,1] x * sqrt(1 + 4x^2) dx. Result: (pi/6)(5*sqrt(5) - 1).

Question 18

y = ln(x) from x = 1 to x = e; rotate about y-axis.

Answer 18

S = 2pi * integral[1,e] x * sqrt(1 + 1/x^2) dx. Result: 2pi * integral sqrt(x^2+1). Result: pi[e*sqrt(e^2+1) + ln(e+sqrt(e^2+1)) - sqrt(2) - ln(1+sqrt(2))].

Question 19

y = sqrt(x) from x = 0 to x = 2; rotate about x-axis.

Answer 19

S = 2pi * integral[0,2] sqrt(x) * sqrt(1 + 1/4x) dx. Result: 13pi/3.

Question 20

y = x^2/2 from x = 0 to x = 1; rotate about x-axis.

Answer 20

S = 2pi * integral[0,1] (x^2/2) * sqrt(1 + x^2) dx. Result: pi/4 [3*sqrt(2) - ln(1+sqrt(2))].

Question 21

x = cos(t), y = sin(t) from t = 0 to t = pi; rotate about x-axis.

Answer 21

S = 2pi * integral[0,pi] sin(t) * 1 dt. Result: 4pi (Surface area of a sphere).

Question 22

y = 1/x from x = 1 to x = 2; rotate about x-axis.

Answer 22

S = 2pi * integral[1,2] (1/x) * sqrt(1 + 1/x^4) dx. (Numerical/Hard integral).