Answer to Question #196306 in Calculus for LF17 Gaming

Question

2- Using a mathematical model and calculus methods (e.g. numerical and

integration methods) to solve given engineering problem (Eq. 1).

Your tasks is

d) Using definite integration and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds.

e) Using a mid-ordinate rule and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds at h= 0.5.

f) Find an accurate mathematical model (e.g. equation) to correlate

position and time. To complete this task you should be able to sketch

the graph again, find the accurate equation using an excel sheet and

trendline.

g) Compare the R2 between the new and previous equations (step c and f).

h) Use the driven equation x (t) from (C) and solve it using the following

numerical methods over the time interval 0 ≤ t ≤ 3 seconds at h=0.5.

I. Using the trapezium method

II. Using a Simpsons rule

Answer 1

(d)Definite integration

    $=\int_0^3 (3t^2+2)dt\\[9pt]=\dfrac{3t^3}{3}+2t|_0^3\\[9pt]=(27-0)+(2(3-0)=33$

(e) $=\int_0^3 (3t^2+2) dt$ ​and h=0.5 assuming n=

The values are-

$\int_0^3 (3t^2+2)dt=\dfrac{h}{2}(2+2.75+5+8.75)\\[9pt] =\dfrac{1.5}{2}\times 18.5 \\[9pt] =13.875$

Hence $\int_0^3 (3t^2+2) dt=13.875$

(f) Mathematical model to correlate position and time-

Distance-

$x(t)=3t^2+2$

(g)The new and previous equation of $R^2$ are same. i.e. $3t^2+2$

(h)

$x(t)=3t^2+2$

(i) Trapezoidal rule-

$=\int_0^3 (3t^2+2) dt$ ​and h=0.5 assuming n=

The values are-

$\int_0^3 (3t^2+2)dt=\dfrac{h}{2}(2+2.75+5+8.75)\\[9pt] =\dfrac{1.5}{2}\times 18.5 \\[9pt] =13.875$

Hence $\int_0^3 (3t^2+2) dt=13.875$

(ii) Using Simpson's rule-

Expression is- $\int_0^3 3t^2+2$

Here, $a=0,b=3, f(t)=3t^2+2$

    $S_x(f):=(f(a)+4f(\dfrac{(a+b)}{2})+f(b)f(\dfrac{(b−a)}{6})$

$=f(0)+4f(1.5)+f(3).f(0.5)\\[9pt]=2+4(3(1.5)^2+2)+(3(3)^2+2)(3(0.5)^2+2)\\[9pt]=2+4(8.75)+(29)(2.75)\\[9pt]=116.75$