#### Top positive review

*4.0 out of 5 stars*A Test of the Speediness of Four Calculators with Numerical Integration

Reviewed in the United States on July 12, 2018

The four calculators are:

1. The TI-36X Pro (scientific calculator)

2. The Casio fx-115ES Plus (scientific calculator)

3. The Casio fx-991EX ClassWiz (scientific calculator)

4. The Casio fx-9750gii (graphing calculator)

The problem used for test:

B = 1.3271244e+20

C = 6.957e+08

D = 1.495978707e+11

Evaluate:

∫ (C, 0.999999D) 1 / √(2B(1/x − 1/D)) dx

This integral returns the amount of time, in seconds, for an object to fall to the sun's photosphere (one solar radius = C) starting from a distance of 1 astronomical unit (D), assuming that the sun and the object are initially at rest with respect to each other and are acted upon by no forces other than their mutual gravitational attraction. The variable B is the solar gravitational parameter.

TI-36X Pro

Answer: 5570898.581

Time to Solve: 90.4 sec

Casio fx-115ES Plus ← this calculator is being reviewed

Answer: 5570898.583

Time to Solve: 76.6 sec

Casio fx-991EX ClassWiz

Answer: 5570898.583

Time to Solve: 17.3 sec

Casio fx-9750gii

Answer: 5570898.583

Time to Solve: 4.9 sec

As you can see, the Texas Instruments device is a slow-poke. The TI-36X Pro costs anywhere from $19 to $30 on Amazon.

The Casio fx-991EX ClassWiz is the official (and substantial) Casio upgrade to the earlier Casio fx-115ES Plus. For moderately complicated numerical integration chores, the Casio fx-991EX is more than four times faster than its predecessor model. Furthermore, the Casio fx-991EX is more than five times faster than its ostensible competitor from Texas Instruments, the TI-36X Pro. The Casio fx-991EX sells at prices ranging from $14 to $36 on Amazon.

But the Casio fx-9750gii is more than three times faster than the Casio fx-991EX ClassWiz and is about 18 times faster than the TI-36X Pro. Reasonable prices on Amazon go from $30 to $37, but there are some sellers (that should be ignored) who are trying to gouge for more.

I have solved the indefinite integral analytically. Omitting the intermediate steps,

∫ 1 / √(2B(1/x − 1/D)) dx

= √[D/(2B)] { √(XD−X²) + D arctan[√(D/X−1)] }

When I used the calculators to solve the integral numerically, I had to avoid entering D itself for X, since that would cause an overflow error. When I use the upper limit of 0.999999D instead, the answer is 5577993.302993855, so all of the calculators' answers have relative errors of about −0.001272.