# Volume of a Solid of Revolution

See how to find the volume of a solid of revolution

(disc method, washer method).

5 examples and their solutions.

## Solid of Revolution

### Definition

formed by rotating a 2D figure (line, curve ...)

around an axis.

## Disc Method

### Formula

V = ∫abπy

^{2}dx

= ∫abπy

^{2}dx

Cross sectional area S(x): πy

^{2}

The shape of πy

^{2}is a disc. (= circle)

→ Disc method

Volume from its Cross Sectional Area

Area of a Circle

V = ∫abπx

^{2}dy

^{2}

### Example

A region is bounded by

y = x

If the region is rotated around the x-axis,

find the volume of the solid of revolution.

Solution y = x

^{4}, x = 1, and the x-axis.If the region is rotated around the x-axis,

find the volume of the solid of revolution.

V = ∫01π⋅(x

^{4})

^{2}dx - [1]

= π∫01x

^{8}dx - [2]

= π[19x

^{9}]01 - [3]

= π9[x

^{9}]01

= π9[1

^{9}- 0

^{9}]

= π9[1 - 0]

= π9

[1]

Close

### Example

A region is bounded by

y = x

If the region is rotated around the y-axis,

find the volume of the solid of revolution.

Solution y = x

^{4}, y = 1, and the y-axis.If the region is rotated around the y-axis,

find the volume of the solid of revolution.

V = ∫01π⋅(y

^{14})

^{2}dy - [1]

= π∫01y

^{12}dy

= π[23x

^{32}]01

= 2π3[x

^{32}]01

= 2π3(1

^{32}- 0

^{32})

= 2π3(1 - 0)

= 2π3

[1]

Close

### Example

Solution [1]

V = ∫-rrπ(r

= π∫-rr(r

= π⋅2∫0r(r

= 2π[r

= 2π[r

= 2π[r

= 2π[3r

= 2π⋅2r

= 43πr

Close

V = ∫-rrπ(r

^{2}- x

^{2}) dx - [2]

= π∫-rr(r

^{2}- x

^{2}) dx

= π⋅2∫0r(r

^{2}- x

^{2}) dx - [3]

= 2π[r

^{2}x - 13x

^{3}]0r

= 2π[r

^{2}⋅r - 13r

^{3}- [r

^{2}⋅0 - 130

^{3}]]

= 2π[r

^{3}- 13r

^{3}- [0 - 0]]

= 2π[3r

^{3}3 - r

^{3}3]

= 2π⋅2r

^{3}3

= 43πr

^{3}

[1]

[2]

x

y

→ S(x) = πy

= π(r

^{2}+ y^{2}= r^{2}y

^{2}= r^{2}- x^{2}→ S(x) = πy

^{2}= π(r

^{2}- x^{2})[3]

r

∫

Definite Integral

^{2}(= x^{0}), -x^{2}: Even functions∫

_{-a}^{a}(even) dx = 2 ∫_{0}^{a}(even) dxDefinite Integral

Close

## Washer Method

### Formula

V = ∫abπ(y

_{1})

^{2}dx - ∫abπ(y

_{2})

^{2}dx

_{1})

^{2}- π(y

_{2})

^{2}

The shape is a washer: ⭗

→ Washer method

### Example

A region is bounded by

y = e

If the region is rotated around the x-axis,

find the volume of the solid of revolution.

Solution y = e

^{x}, the tangent of y = e^{x}at (1, e), and the y-axis.If the region is rotated around the x-axis,

find the volume of the solid of revolution.

f(x) = e

f'(x) = e

f'(1) = e

= e

(1, e)

y = e(x - 1) + e - [2]

= ex - e + e

y = ex

V = ∫01π(e

= π∫01e

= π∫01e

= π[12e

= π2[e

= π2[e

= π2[e

= π2[e

= πe

= 3πe

= 3πe

= πe

= π(e

^{x}f'(x) = e

^{x}- [1]f'(1) = e

^{1}= e

(1, e)

y = e(x - 1) + e - [2]

= ex - e + e

y = ex

V = ∫01π(e

^{x})^{2}dx - ∫01π(ex)^{2}dx - [3]= π∫01e

^{2x}dx - π∫01e^{2}x^{2}dx - [4]= π∫01e

^{2x}dx - πe^{2}∫01x^{2}dx= π[12e

^{2x}]01 - πe^{2}[13x^{3}]01 - [5]= π2[e

^{2x}]01 - πe^{2}3[x^{3}]01= π2[e

^{2⋅1}- e^{2⋅0}] - πe^{2}3[1^{3}- 0^{3}]= π2[e

^{2}- e^{0}] - πe^{2}3[1 - 0]= π2[e

^{2}- 1] - πe^{2}3= πe

^{2}2 - π2 - πe^{2}3= 3πe

^{2}6 - 3π6 - 2πe^{2}6= 3πe

^{2}- 3π - 2πe^{2}6= πe

^{2}- 3π6= π(e

^{2}- 3)6[2]

Slope of the tangent line: e

Tangent line passes through (1, e).

→ Tangent line: y = e(x - 1) + e

Equation of a Tangent Line (Derivative)

Tangent line passes through (1, e).

→ Tangent line: y = e(x - 1) + e

Equation of a Tangent Line (Derivative)

[3]

Outer function: y = e

Inner function: y = ex

^{x}Inner function: y = ex

Close

### Example

A region is bounded by

y = 2√x - 1, the tangent of the function at (2, 2), and the x-axis.

If the region is rotated around the x-axis,

find the volume of the solid of revolution.

Solution y = 2√x - 1, the tangent of the function at (2, 2), and the x-axis.

If the region is rotated around the x-axis,

find the volume of the solid of revolution.

f(x) = 2(x - 1)

f'(x) = 2⋅12⋅(x - 1)

= (x - 1)

f'(2) = (2 - 1)

= 1

= 1

(2, 2)

y = 1(x - 2) + 2 - [2]

y = x

V = ∫02πx

= π∫02x

= π∫02x

= π[13x

= π[13⋅2

= π[13⋅8 - 0] - 4π[12⋅4 - 2 - [12⋅1 - 1]]

= π⋅83 - 4π[2 - 2 - [12 - 1]]

= 8π3 - 4π[-[12 - 22]]

= 8π3 + 4π[-12]

= 8π3 - 2π

= 8π3 - 6π3

= 2π3

^{12}f'(x) = 2⋅12⋅(x - 1)

^{-12}⋅1 - [1]= (x - 1)

^{-12}f'(2) = (2 - 1)

^{-12}= 1

^{-12}= 1

(2, 2)

y = 1(x - 2) + 2 - [2]

y = x

V = ∫02πx

^{2}dx - ∫12π⋅[2(x - 1)^{12}]^{2}dx - [3]= π∫02x

^{2}dx - π∫124(x - 1) dx= π∫02x

^{2}dx - 4π∫12(x - 1) dx= π[13x

^{3}]02 - 4π[12x^{2}- x]12= π[13⋅2

^{3}- 13⋅0^{3}] - 4π[12⋅2^{2}- 2 - [12⋅1^{2}- 1]]= π[13⋅8 - 0] - 4π[12⋅4 - 2 - [12⋅1 - 1]]

= π⋅83 - 4π[2 - 2 - [12 - 1]]

= 8π3 - 4π[-[12 - 22]]

= 8π3 + 4π[-12]

= 8π3 - 2π

= 8π3 - 6π3

= 2π3

[2]

Slope of the tangent line: 1

Tangent line passes through (2, 2).

→ Tangent line: y = 1(x - 2) + 2

Tangent line passes through (2, 2).

→ Tangent line: y = 1(x - 2) + 2

[3]

Outer function: y = x

Inner function: y = 2(x - 1)

Inner function: y = 2(x - 1)

^{1/2}Close