# Integration Definitions, Properties & Theorems

Lead Author(s): **Kelly Cline**

Student Price: **FREE**

These questions explore integration defined through Riemann sums, as well as basic properties and theorems of integrals

Content Index

[Theorems] 1

If $f(x) = \int_1^x t^4 dt$ , then

A

$f ’ (x) = t^4$

B

$f ’ (x) = x^4$

C

$f ’ (x) = \frac{1}{5} x^5 – \frac{1}{5}$

D

$f ’ (x) = x^4 – 1$

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[Riemann sums] 1

$\bf{True or False:}$ If a piece of string has been chopped into $n$ small pieces and the $i^{th}$ piece is $\Delta x_i$ inches long, then the total length of the string is exactly $\sum_{i=1}^n \Delta x_i$ .

A

True, and I am very confident

B

True, but I am not very confident

C

False, but I am not very confident

D

False, and I am very confident

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