Exercises

1. Consider the area under the curve $y=\sqrt{4-x^2}$ on the interval $[-2,2]$.
   1. Use the formula for the area of a circle to compute the exact value of this area.
   2. Using the left endpoint rule, find the minimum number of subintervals required to approximate this area to within a tolerance of $0.05$. That is find the value of $n$ such that $\mid A- L_n \mid < 0.05$, but $\mid A- L_{n-1} \mid > 0.05$.

2. Consider the function
   
   $$g(x) = \frac{x^3-5x^2-26x+40}{x^2+5}$$
   
   
   on the interval $[-4,1]$.
   1. Use the error bound formula to find the smallest value of $n$ that guarantees that $M_n$ approximates the area to within $0.001$. That is, find the smallest value of $n$ that guarantees that $\mid EM_n \mid < 0.001$.

   2. The value of $n$ given by the error bound is usually conservative. That is, in practice the desired accuracy can be achieved with a smaller value of $n$. Given that
      
      $$\int_{-4}^1 g(x) \, dx \approx 29.98602$$
      
      
      find the smallest value of $n$ such that $\mid EM_n \mid < 0.001$ and compare it to the value you obtained in the previous exercise.

3. Repeat the previous exercise, but use the left endpoint rule. That is, find the smallest value of $n$ such that
   
   $$\mid L_n - 29.98602 \mid < 0.001$$
   
   
   Your value of $n$ should be much larger than the one in the previous exercise.

4. The midpoint rule is usually a much better approximation to the area than either the left endpoint or right endpoint rules. However, suppose you approximate the area by taking the average of the values given by the left endpoint or right endpoint rules. That is, we define a new approximation $T_n$ by
   
   $$T_n = (L_n + R_n)/2$$
   
   
   Using the same function $g(x)$ and the same interval $-4 \leq x \leq 1$, find the smallest value of $n$ such that
   
   $$\mid T_n - 29.98602 \mid < 0.001$$
   
   
   Is this value of $n$ closer to the one you found for the midpoint rule or the one you found for the left endpoint rule? Does this make sense to you?