The total mass of a variable density rod is given by m = L 0 (x)Ac(x) dx where m = mass

Derive Eq. (19.4) by integrating Eq. (19.3).

Evaluate the following integral: 4 0 (1 ex ) dx (a) analytically, (b) single application of the trapezoidal rule, (c) composite trapezoidal rule with n = 2 and 4, (d) single application of Simpsons 1/3 rule, (e) composite Simpsons 1/3 rule with n = 4, (f) Simpsons 3/8 rule, and (g) composite Simpsons rule, with n = 5. For each of the numerical estimates (b) through (g), determine the true percent relative error based on (a).

Evaluate the following integral: /2 0 (8 + 4 cos x) dx (a) analytically, (b) single application of the trapezoidal rule, (c) composite trapezoidal rule with n = 2 and 4, (d) single application of Simpsons 1/3 rule, (e) composite Simpsons 1/3 rule with n = 4, (f) Simpsons 3/8 rule, and (g) composite Simpsons rule, with n = 5. For each of the numerical estimates (b) through (g), determine the true percent relative error based on (a).

Evaluate the following integral: 4 2 (1 x 4x3 + 2x5) dx (a) analytically, (b) single application of the trapezoidal rule, (c) composite trapezoidal rule with n = 2 and 4, (d)single application of Simpsons 1/3 rule, (e) Simpsons 3/8 rule, and (f)Booles rule. For each of the numerical estimates(b)through (f), determine the true percent relative error based on (a).

The function f (x) = ex can be used to generate the following table of unequally spaced data: x 0 0.1 0.3 0.5 0.7 0.95 1.2 f (x) 1 0.9048 0.7408 0.6065 0.4966 0.3867 0.3012 Evaluate the integral from a = 0 to b = 0.6 using (a) analytical means, (b) the trapezoidal rule, and (c) a combination of the trapezoidal and Simpsons rules wherever possible to attain the highest accuracy. For (b) and (c), compute the true percent relative error.

Evaluate the double integral 2 2 4 0 (x2 3y2 + xy3) dx dy (a) analytically, (b) using the composite trapezoidal rule with n = 2, and (c) using single applications of Simpsons 1/3 rule. For (b) and (c), compute the percent relative error. 0

Evaluate the triple integral 4 4 6 0 3 1 (x3 2yz) dx dy dz (a) analytically, and (b) using single applications of Simpsons 1/3 rule. For (b), compute the true percent relative error. 1

Determine the distance traveled from the following velocity data: t 1 2 3.25 4.5 6 7 8 8.5 9 10 v 5 6 5.5 7 8.5 8 6 7 7 5 (a) Use the trapezoidal rule. In addition, determine the average velocity. (b) Fit the data with a cubic equation using polynomial regression. Integrate the cubic equation to determine the distance.

Water exerts pressure on the upstream face of a dam as shown in Fig. P19.9. The pressure can be characterized by p(z) = g(D z) where p(z) = pressure in pascals (or N/m2) exerted at an elevation z meters above the reservoir bottom; = density of water, which for this problem is assumed to be a constant 103 kg/m3; g = acceleration due to gravity (9.81 m/s2); and D = elevation (in m) of the water surface above the reservoir bottom. According to Eq. (P19.9), pressure increases linearly with depth, as depicted in Fig. P19.9a. Omitting atmospheric pressure (because it works against both sides of the dam face and essentially cancels out), the total force ft can be determined by multiplying pressure times the area of the dam face (as shown in Fig. P19.9b). Because both pressure and area vary with elevation, the total force is obtained by evaluating ft = D 0 gw(z)(D z) dz where w(z) = width of the dam face (m) at elevation z (Fig. P19.9b). The line of action can also be obtained by evaluating d =  D 0 gzw(z)(D z) dz  D 0 gw(z)(D z) dz Use Simpsons rule to compute ft and d

A wind force distributed against the side of a skyscraper is measured as Height l, m 0 30 60 90 120 Force, F(l), N/m 0 340 1200 1550 2700 Height l, m 150 180 210 240 Force, F(l), N/m 3100 3200 3500 3750 Compute the net force and the line of action due to this distributed wind.

An 11-m beam is subjected to a load, and the shear force follows the equation V(x) = 5 + 0.25x2 where V is the shear force, and x is length in distance along the beam. We know that V = dM/dx, and M is the bending moment. Integration yields the relationship M = Mo + x 0 V dx I If Mo is zero and x = 11, calculate M using (a) analytical integration, (b) composite trapezoidal rule, and (c) composite Simpsons rules. For (b) and (c) use 1-m increments.

The total mass of a variable density rod is given by m = L 0 (x)Ac(x) dx where m = mass, (x) = density, Ac(x) = cross-sectional area, x = distance along the rod and L = the total length of the rod. The following data have been measured for a 20-m length rod. Determine the mass in grams to the best possible accuracy.

A transportation engineering study requires that you determine the number of cars that pass through an intersection traveling during morning rush hour. You stand at the side of the road and count the number of cars that pass every 4 minutes at several times as tabulated below. Use the best numerical method to determine (a) the total number of cars that pass between 7:30 and 9:15, and (b) the rate of cars going through the intersection per minute. (Hint: Be careful with units.)

5 Determine the average value for the data in Fig. P19.15. Perform the integral needed for the average in the order shown by the following equation: I = xn x0  ym y0 f (x, y) dy dx 19

Integration provides a means to compute how much mass enters or leaves a reactor over a specified time period, as in M = t2 t1 Qc dt where t1 and t2 = the initial and final times, respectively. This formula makes intuitive sense if you recall the analogy between integration and summation. Thus, the integral represents the summation of the product of flow times concentration to give the total mass entering or leaving from t1 to t2. Use numerical integration to evaluate this equation for the data listed below:

The cross-sectional area of a channel can be computed as Ac = B 0 H(y) dy where B = the total channel width (m), H = the depth (m), and y = distance from the bank (m). In a similar fashion, the average flow Q (m3 /s) can be computed as Q = B 0 U(y)H(y) dy where U = water velocity (m/s). Use these relationships and a numerical method to determine Ac and Q for the following data:

The average concentration of a substance c(g/m3 ) in a lake where the area As(m2 ) varies with depth z(m) can be computed by integration: c =  Z 0 c(z)As(z) dz  Z 0 As(z) dz where Z = the total depth (m). Determine the average concentration based on the following data:

As was done in Section 19.9, determine the work performed if a constant force of 1 N applied at an angle results in the following displacements. Use the cumtrapz function to determine the cumulative work and plot the result versus .

As specified in the following table, a manufactured spherical particle has a density that varies as a function of the distance from its center (r = 0): r, mm 0 0.12 0.24 0.36 0.49 (g/cm3) 6 5.81 5.14 4.29 3.39 r, mm 0.62 0.79 0.86 0.93 1 (g/cm3) 2.7 2.19 2.1 2.04 2 Use numerical integration to estimate the particles mass (in g) and average density (in g/cm3 ).

As specified in the following table, the earths density varies as a function of the distance from its center (r 0): r, km 0 1100 1500 2450 3400 3630 (g/cm3) 13 12.4 12 11.2 9.7 5.7 r, km 4500 5380 6060 6280 6380 (g/cm3) 5.2 4.7 3.6 3.4 3 Use numerical integration to estimate the earths mass (in metric tonnes) and average density (in g/cm3 ). Develop vertically stacked subplots of (top) density versus radius, and (bottom) mass versus radius. Assume that the earth is a perfect sphere.