Let ƒ be an integrable function on [a,b]. Which of the following is always true?

A) $\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )$
B) $\int _ { a } ^ { b } f ( x ) d x = \int _ { a / 2 } ^ { b / 2 } 2 f ( x ) d x$
C) $\int _ { a } ^ { b } f ( x ) d x = \frac { 1 } { 2 } \int _ { 2 a } ^ { 2 b } f ( x ) d x$
D) $\int _ { a } ^ { b } f ( x ) d x 0$
E)is bounded. $\int _ { a } ^ { b } f ( x ) d x$

Question

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

A) $\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )$
B) $\int _ { a } ^ { b } f ( x ) d x = \int _ { a / 2 } ^ { b / 2 } 2 f ( x ) d x$
C) $\int _ { a } ^ { b } f ( x ) d x = \frac { 1 } { 2 } \int _ { 2 a } ^ { 2 b } f ( x ) d x$
D) $\int _ { a } ^ { b } f ( x ) d x > 0$
E)is bounded. $\int _ { a } ^ { b } f ( x ) d x$

Quizplus · Accepted Answer

The integral of a function over a closed interval [a, b] is always a finite number if the function is integrable on that interval, which means the integral is bounded. Choices A, B, C, and D make specific claims about the value of the integral that are not universally true for all integrable functions or their integrals.

Let ƒ Be an Integrable Function on [A,b] ∫abf(x)dx=f(x)(b−a)\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )∫ab​f(x)dx=f(x)(b−a)

Let ƒ Be an Integrable Function on [A,b] $\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )$