Let ƒ be an integrable function on [a,b]. Which of the following is always true? A) $\int _ { a } ^ { b } f ( x ) d x \int _ { a } ^ { b } f ( a ) d x$ C) $\int _ { a } ^ { b } f ( x ) d x \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$ D)is a real number. $\int _ { a } ^ { b } f ( x ) d x$ E) $\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) $\int _ { a } ^ { b } f ( x ) d x < \int _ { a } ^ { b } f ( b ) d x$ B) $\int _ { a } ^ { b } f ( x ) d x > \int _ { a } ^ { b } f ( a ) d x$ C) $\int _ { a } ^ { b } f ( x ) d x > \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$ D)is a real number. $\int _ { a } ^ { b } f ( x ) d x$ E) $\int _ { a } ^ { b } f ( x ) d x < a f ( b )$

Quizplus · Accepted Answer

The integral of an integrable function over a closed interval [a,b] is always a real number, which is a fundamental property of definite integrals. Choices A, B, C, and E make specific claims about the relationship between the integral and the function values or constants that are not universally true without additional information about the function $f$.

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

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