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Find the approximations $L_{n}, R_{n}, T_{n},$ and $M_{n}$ to the integral$\int_{0}^{1} x e^{x} d x$ for $n=5,10,$ and $20 .$ Then compute the corresponding errors $E_{L}, E_{R}, E_{T},$ and $E_{M}$ . (Round your answersto six decimal places. You may wish to use the sum command on a computer algebra system.) What observationscan you make? In particular, what happens to the errorswhen $n$ is doubled?

$$L_{5} \approx 0.742943$$$$R_{5} \approx 1.286599$$$$T_{5} \approx 1.014771$$$$M_{5} \approx 0.992621$$$$\begin{array}{l}{E_{L}=1-0.742943} \\ {=0.257057}\end{array}$$$$\begin{array}{l}{E_{R}=1-1.286599} \\ {=-0.286599}\end{array}$$$$\begin{array}{l}{E_{T}=1-1.014771} \\ {=-0.014771}\end{array}$$$$\begin{array}{l}{E_{M}=1-0.992621} \\ {=0.007379}\end{array}$$

Calculus 2 / BC

Chapter 6

TECHNIQUES OF INTEGRATION

Section 5

Approximate Integration

Integration Techniques

Improper Integrals

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