An integral is a mathematical operation that involves finding the area under a curve. It can be used to calculate various quantities, such as the total distance traveled, the volume of a solid, the work done by a force, and more.
The TI-84 calculator has a built-in function that can compute integrals for you, as long as the function is continuous and the limits of integration are finite. In this article, we will show you how to use the integral function on a TI-84 calculator, and provide some examples and tips along the way.
How to Access the Integral Function
The integral function can be accessed on a TI-84 calculator by following these steps:
- Press the MATH key and then press the right-arrow key to move to the MATH menu.
- Scroll down to option 9, or just press 9, to select fnInt(. This will bring up the integral function on the screen.
How to Use the Integral Function
The integral function has the following syntax:
fnInt(function, variable, lower limit, upper limit)
where:
- function: the function you want to integrate, using the appropriate symbols and parentheses
- variable: the variable of integration, using the letter X
- lower limit: the lower bound of the interval of integration
- upper limit: the upper bound of the interval of integration
The integral function will return the value of the integral of the function over the interval.
Examples of Using the Integral Function
Let’s look at some examples of how to use the integral function on a TI-84 calculator.
Example 1: Simple Integral
Question: Find the value of the integral of x2 from 0 to 3.
Answer: To find the value of the integral, we need to use the integral function with the following arguments:
fnInt(X2,X,0,3)
Press ENTER to get the answer: 9
This means that the integral of x2 from 0 to 3 is equal to 9.
Example 2: Integral with Trigonometric Function
Question: Find the value of the integral of sin(x) from 0 to pi.
Answer: To find the value of the integral, we need to use the integral function with the following arguments:
fnInt(sin(X),X,0,pi)
Note: To enter the pi symbol, press 2nd and then press the x-1 key, which is located above the [x2] key.
Press ENTER to get the answer: 2
This means that the integral of sin(x) from 0 to pi is equal to 2.
Example 3: Integral with Exponential Function
Question: Find the value of the integral of ex from 0 to 1.
Answer: To find the value of the integral, we need to use the integral function with the following arguments:
fnInt(eX,X,0,1)
Note: To enter the e symbol, press 2nd and then press the ln key, which is located above the [x-1] key.
Press ENTER to get the answer: 1.718282
This means that the integral of ex from 0 to 1 is approximately equal to 1.718282.
Tips and Tricks for Using the Integral Function
Here are some tips and tricks for using the integral function on a TI-84 calculator:
- To enter the function, you can use the following symbols and keys on the calculator:
- X: the variable key, located above the STO> key
- +: the plus sign, located next to the ENTER key
- -: the minus sign, located next to the plus sign
- x: the multiplication sign, located above the plus sign
- /: the division sign, located above the minus sign
- ^: the exponent sign, located above the division sign
- (: the left parenthesis, located above the 8 key
- ): the right parenthesis, located above the 9 key
- sin: the sine function, located above the 7 key
- cos: the cosine function, located above the 8 key
- tan: the tangent function, located above the 9 key
- e: the natural base, located above the ln key
- pi: the mathematical constant, located above the x-1 key
- To enter the limits of integration, you can use any number or expression that is valid on the calculator. For example, you can use fractions, decimals, radicals, or other functions. To enter a fraction, press the ALPHA key and then press the Y= key. This will bring up the fraction template on the screen. To enter a radical, press the 2nd key and then press the x2 key. This will bring up the radical symbol on the screen. To enter other functions, press the MATH key and then scroll through the menus to find the function you want.
- To find the value of an improper integral, where one or both of the limits of integration are infinite, you can use a very large or very small number to approximate the infinity symbol. For example, to find the value of the integral of 1/x from 1 to infinity, you can use the integral function with the following argument:
fnInt(1/X,X,1,1E99)
Note: To enter the 1E99, press 1 EE 99. This will display 1E99 on the screen, which means 1 x 1099.
Press ENTER to get the answer: 9.9999999E98
This means that the integral of 1/x from 1 to infinity is approximately equal to 9.9999999 x 1098.
- To check your answer, you can use the antiderivative function on the calculator. The antiderivative function can be accessed by pressing the MATH key and then scrolling down to option A:antideriv(. The antiderivative function has the following syntax:
antideriv(function, variable)
where:
- function: the function you want to find the antiderivative of, using the appropriate symbols and parentheses
- variable: the variable of integration, using the letter X
The antiderivative function will return the antiderivative of the function, or an error message if the antiderivative cannot be found.
To check your answer, you can use the antiderivative function to find the antiderivative of the function, and then evaluate it at the limits of integration using the STO> key. For example, to check the answer for the integral of x2 from 0 to 3, you can use the antiderivative function with the following argument:
antideriv(X2,X)
Press ENTER to get the answer:
X3/3
This means that the antiderivative of x2 is x3/3. To evaluate it at the limits of integration, you can press:
2nd ENTER 3 STO> X ENTER
The calculator will display:
X3/3
9
This means that the antiderivative of x2 evaluated at 3 is 9. To find the difference between the antiderivative evaluated at the upper and lower limits, you can press:
- 2nd ENTER 0 STO> X ENTER
The calculator will display:
X3/3
0
This means that the antiderivative of x2 evaluated at 0 is 0. To find the difference, you can press:
- ANS ENTER
The calculator will display:
9
This means that the difference between the antiderivative evaluated at the upper and lower limits is 9, which is the same as the value of the integral.