Let on the interval

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

g(x)=\ln x

on the interval

[1,3]

. Using two equal subintervals, a student computes the midpoint approximation

M

and the trapezoidal approximation

T

\displaystyle \int_{1}^{3} g(x)\,dx

. Without evaluating the exact integral, which of the following statements is true? (a)

T<M<\displaystyle\int_{1}^{3} g(x)\,dx

(b)

\displaystyle\int_{1}^{3} g(x)\,dx<T<M

(c)

T<\displaystyle\int_{1}^{3} g(x)\,dx<M

and

\left|M-\int_{1}^{3} g(x)\,dx\right|<\left|T-\int_{1}^{3} g(x)\,dx\right|

(d)

T<\displaystyle\int_{1}^{3} g(x)\,dx<M

and

\left|M-\int_{1}^{3} g(x)\,dx\right|>\left|T-\int_{1}^{3} g(x)\,dx\right|