Try the fastest way to create flashcards
Question

# A draftsman is asked to determine the amount of material required to produce a machine part. The diameters d of the part at equally spaced points x are listed in the table. The measurements are listed in centimeters. x 0 1 2 3 4 5 d 4.2 3.8 4.2 4.7 5.2 5.7 x 6 7 8 9 10 d 5.8 5.4 4.9 4.4 4.6 Use these data with Simpson's Rule to approximate the volume of the part.

Solution

Verified
Step 1
1 of 3

We know that the Simpson's Rule for area looks like:

$\int_a^b f(x) \ dx \approx \dfrac{\Delta x}{3} \left( f(x_0) + 4 f(x_1) + 2 f(x_2) + \ldots + 4 f(x_{n-1}) + f(x_n) \right)$

Since the volume of a rotate region is calculated by $V = \displaystyle \pi \int_a^b [f(x)]^2 \ dx$, Simpson's Rule for volume looks like:

$\pi \int_a^b [f(x)]^2 \ dx \approx \dfrac{\Delta x \cdot \pi}{3} \left( f(x_0)^2 + 4 f(x_1)^2 + 2 f(x_2)^2 + \ldots + 4 f(x_{n-1})^2 + f(x_n)^2 \right)$

## Recommended textbook solutions

#### Thomas' Calculus

14th EditionISBN: 9780134438986 (11 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,142 solutions

#### Calculus of a Single Variable

6th EditionISBN: 9780395885789Bruce H. Edwards, Robert P. Hostetler, Ron Larson
7,505 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550James Stewart
11,083 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (2 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions