## Related questions with answers

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

VerifiedWe 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:

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

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