Mathematics

Mathematics

Problem

Let g(x)=ln⁡xg(x)=\ln x on the interval [1,3][1,3]. Using two equal subintervals, a student computes the midpoint approximation MM and the trapezoidal approximation TT to ∫13g(x) dx\displaystyle \int_{1}^{3} g(x)\,dx. Without evaluating the exact integral, which of the following statements is true? a) T<M<∫13g(x) dxT<M<\displaystyle\int_{1}^{3} g(x)\,dx b) ∫13g(x) dx<T<M\displaystyle\int_{1}^{3} g(x)\,dx<T<M c) T<∫13g(x) dx<MT<\displaystyle\int_{1}^{3} g(x)\,dx<M and ∣M−∫13g(x) dx∣<∣T−∫13g(x) dx∣\left|M-\int_{1}^{3} g(x)\,dx\right|<\left|T-\int_{1}^{3} g(x)\,dx\right| d) T<∫13g(x) dx<MT<\displaystyle\int_{1}^{3} g(x)\,dx<M and ∣M−∫13g(x) dx∣>∣T−∫13g(x) dx∣\left|M-\int_{1}^{3} g(x)\,dx\right|>\left|T-\int_{1}^{3} g(x)\,dx\right|

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