Numerical integration

In this activity, we explore the relationships among the errors generated by left, right, midpoint, and trapezoid approximations to the definite integral $\int_1^2 \frac{1}{x^2} \, dx$.

1. Use the First FTC to evaluate $\int_1^2 \frac{1}{x^2} \, dx$ exactly.

2. Use appropriate computing technology to compute the following approximations for $\int_1^2 \frac{1}{x^2} \, dx$: $T_4$, $M_4$, $T_8$, and $M_8$.

3. Let the error that results from an approximation be the approximation's value minus the exact value of th…

Consider the definite integral $\int_0^1 x \tan(x) \, dx$.

1. Explain why this integral cannot be evaluated exactly by using either $u$-substitution or by integrating by parts.

2. Using appropriate subintervals, compute $L_4$, $R_4$, $M_4$, $T_4$, and $S_8$.

3. 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 $M_4 = 7.05$.

1. Is $f$ increasing or decreasing on $[3,6]$? What data tells you?

2. Is $f$ concave up or concave down on $[3,6]$? Why?

3. Determine the best possible estimate you can for $\int_3^6 f(x) \, dx$, based on the given infor…

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.

1. 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?

2. Use the given data to calculate $M_n$ for…