Examine the integral $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x$.

1. Explain why this integral cannot be evaluated exactly.

2. Estimate $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x$, using a Riemann sum with five left endpoint rectangles.

   Step (b):

   $0.2e^0+0.2e^{-0.2^2}+0.2e^{-0.4^2}+0.2e^{-0.6^2}+0.2e^{-0.8^2}$

3. Use substitution and the sixth-degree Maclaurin polynomial for $f(x) = e^x$ to get an approximation of $f ( x ) = e ^ { - x ^ { 2 } }$.

   Hint (c):

   $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$

4. Estimate $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x$ using your polynomial from part (c).

   Step (d):

   $\int_0^1\Big(1-x^2+\frac{x^4}{2}-\frac{x^6}{6}\Big)dx$

5. How can you improve your answers to parts (b) and (d)?

   Partial answer (e):

   Use more ________.