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

1. Explain why this integral cannot be evaluated exactly using any of our integration tools.

2. Estimate $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x$, using a Riemann sum with $5$ left-endpoint rectangles.

3. Use a substitution in the expansion of $f(x) = e^x$ to get a $6$^th degree polynomial 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 the polynomial of part (c).

   Hint (d):

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

5. How could we have improved our answers to parts (b) and (d)?

   Partial answer (e):

   Use more ________.