§5.6 Numerical integration (continued)

Overall observations regarding

Summary

On the three sets of axes in the provided figure, sketch a graph of each function on the interval $[0,1]$, and compute $L_1$ and $R_1$ for each. What do you observe?

Compute $M_1$ for each function to approximate $\int_0^1 f(x) \,dx$, $\int_0^1 g(x) \,dx$, and $\int_0^1 h(x) \,dx$, respectively.

Compute $T_1$ for each of the three functions, and hence compute $S_2$ for each of the three functions.

Evaluate each of the integrals $\int_0^1 f(x) \,dx$, $\int_0^1 g(x) \,dx$, and $\int_0^1 h(x) \,dx$ exactly using the First FTC.

For each of the three functions $f$, $g$, and $h$, compare the results of $L_1$, $R_1$, $M_1$, $T_1$, and $S_2$ to the true value of the corresponding definite integral. What patterns do you observe?

Consider the definite integral $\int_0^1 x \tan(x) \, dx$. - Explain why this integral cannot be evaluated exactly by using either $u$-substitution or by integrating by parts. - Using appropriate subintervals, compute $L_4$, $R_4$, $M_4$, $T_4$, and $S_8$. - Which of the approximations in (b) is an over-estimate to the true value of $\int_0^1 x \tan(x) \, dx$? Which is an under-estimate? How do you know?

For an unknown function $f(x)$, the following information is known. -$f$ is continuous on $[3,6]$; -$f$ is either always increasing or always decreasing on $[3,6]$; -$f$ has the same concavity throughout the interval $[3,6]$; - As approximations to $\int_3^6 f(x) \, dx$, $L_4 = 7.23$, $R_4 = 6.75$, and $M_4 = 7.05$. - Is $f$ increasing or decreasing on $[3,6]$? What data tells you? - Is $f$ concave up or concave down on $[3,6]$? Why? - Determine the best possible estimate you can for $\int_3^6 f(x) \, dx$, based on the given information.

The rate at which water flows through Table Rock Dam on the White River in Branson, MO, is measured in cubic feet per second (CFS). As engineers open the floodgates, flow rates are recorded according to the following chart.

seconds, $t$ | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ | $60$

flow in CFS, $r(t)$ | $2000$ | $2100$ | $2400$ | $3000$ | $3900$ | $5100$ | $6500$

• What definite integral measures the total volume of water to flow through the dam in the 60 second time period provided by the table above? - Use the given data to calculate $M_n$ for the largest possible value of $n$ to approximate the integral you stated in (a). Do you think $M_n$ over- or under-estimates the exact value of the integral? Why? - Approximate the integral stated in (a) by calculating $S_n$ for the largest possible value of $n$, based on the given data. - Compute $\frac{1}{60} S_n$ and $\frac{2000+2100+2400+3000+3900+5100+6500}{7}$. What quantity do both of these values estimate? Which is a more accurate approximation?