Estimate $\int_{0}^{\,1} {\cos(x^2)}\, dx$ using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with $n = 4$. Give each answer correct to five decimal places.

(a) $T_4 =$

(b) $M_4 =$

(c) By looking at a sketch of the graph of the integrand, determine for each estimate whether it overestimates, underestimates, or is the exact area.

? Underestimate Overestimate Exact  1. $T_4$

? Underestimate Overestimate Exact  2. $M_4$

(d) What can you conclude about the true value of the integral?

A. $T_4 < \int_{0}^{\,1} {\cos(x^2)}\, dx < M_4$
B. $T_4 < \int_{0}^{\,1} {\cos(x^2)}\, dx$ and $M_4 < \int_{0}^{\,1} {\cos(x^2)}\, dx$
C. No conclusions can be drawn.
D. $T_4 > \int_{0}^{\,1} {\cos(x^2)}\, dx$ and $M_4 > \int_{0}^{\,1} {\cos(x^2)}\, dx$
E. $M_4 < \int_{0}^{\,1} {\cos(x^2)}\, dx < T_4$