Arc Length and Surface Area
Department of Mathematics and Statistics
February 21, 2012
Calculus II (James Madison University)
Math 236
February 21, 2012
1/4
Arc Length
Theorem
Suppose f (x) is a continuous, differentiable function with a continuous
derivative. The arc length of f (x) from x = a to x = b can be represented
by the definite integral:
!
a
Calculus II (James Madison University)
b
"
1 + (f ! (x))2 dx.
Math 236
February 21, 2012
2/4
Arc Length
Theorem
Suppose f (x) is a continuous, differentiable function with a continuous
derivative. The arc length of f (x) from x = a to x = b can be represented
by the definite integral:
!
a
Calculus II (James Madison University)
b
"
1 + (f ! (x))2 dx.
Math 236
February 21, 2012
2/4
Arc Length
Theorem
Suppose f (x) is a continuous, differentiable function with a continuous
derivative. The arc length of f (x) from x = a to x = b can be represented
by the definite integral:
!
a
Calculus II (James Madison University)
b
"
1 + (f ! (x))2 dx.
Math 236
February 21, 2012
2/4
Frustums of Cones
Definition
A frustum is a truncated cone.
Theorem
A frustum with radii p and q and slant length s has surface area π(p + q)s.
Calculus II (James Madison University)
Math 236
February 21, 2012
3/4
Frustums of Cones
Definition
A frustum is a truncated cone.
Theorem
A frustum with radii p and q and slant length s has surface area π(p + q)s.
Calculus II (James Madison University)
Math 236
February 21, 2012
3/4
Frustums of Cones
Definition
A frustum is a truncated cone.
Theorem
A frustum with radii p and q and slant length s has surface area π(p + q)s.
Calculus II (James Madison University)
Math 236
February 21, 2012
3/4
Surface Area
Theorem
Suppose f (x) is a differentiable function with a continuous derivative and
that S is the solid of revolution obtained by revolving f around the x-axis.
The surface area of S from x = a to x = b is
! b
"
2π
f (x) 1 + (f ! (x))2 dx.
a
Calculus II (James Madison University)
Math 236
February 21, 2012
4/4
Surface Area
Theorem
Suppose f (x) is a differentiable function with a continuous derivative and
that S is the solid of revolution obtained by revolving f around the x-axis.
The surface area of S from x = a to x = b is
! b
"
2π
f (x) 1 + (f ! (x))2 dx.
a
Calculus II (James Madison University)
Math 236
February 21, 2012
4/4