If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

It is the **most powerful prayer**.
A **pure heart**, a **clean mind**, and a **clear conscience** is necessary for it.

For in GOD we live, and move, and have our being.

- Acts 17:28

The

- Samuel Dominic Chukwuemeka**Joy** of a **Teacher** is the **Success** of his **Students.**

Home
Other Calculators

Integrals and Other Applications

Area Between Curves
Trapezoidal Rule and Simpson's Rule

I greet you this day,

Please solve the questions on your own before checking your answers with the calculators.

I wrote the codes for some of the calculators using JavaScript, a client-side scripting language.

The Wolfram Alpha widgets (many thanks to the developers) were used for some calculators.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.

If you are my student, please do not contact me here. Contact me via the school's system.

Thank you for visiting.

**Samuel Dominic Chukwuemeka** (Samdom For Peace)
B.Eng., A.A.T, M.Ed., M.S

Use only $x$ and $y$

$x$ is the independent vavariable.

$y$ is the dependent variable.

Use $e$ and $h$ appropriately/accordingly

__This calculator will:__

(1.) Determine the antiderivative of a function (indefinite integral of the function).

(2.) Graph the antiderivative of the function.

__To use the calculator, please:__

(1.) Type the function in the textbox (the bigger textbox).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed, into the small textbox of the calculator.

(5.) Click the "Submit" button.

(6.) **Check to make sure that it is the correct function you typed.**

(7.) Review the answers. At least one of the answers is probably what you need.

- Using the Indefinite Integrals Calculator
- Type: $7$
__as__7 - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $y = 4x + 3$
__as__y = 4 * x + 3 - Type: $4x^3 - 5x^{\dfrac{1}{2}} + 7x^{-\dfrac{2}{3}}$
__as__4 * x^3 - 5 * x^(1/2) + 7 * x^(-2/3) - Type: $y = 4x^3 - 5x^2 + 4$
__as__y = 4 * x^2 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $y = |-7 - 5x|$
__as__y = |-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $y = \sec^2 x$
__as__y = sec^2 x - Type: $y = \cos hx$
__as__y = cos hx - Type: $y = \dfrac{1}{1 - x^2}$
__as__y = 1 / (1 - x^2) - Type: $y = \dfrac{-1}{\sqrt{1 - x^2}}$
__as__y = -1 / (sqrt(1 - x^2))

Integrate $wrt\;x$

Use only $x$ and $y$

$x$ is the independent vavariable.

$y$ is the dependent variable.

Use $e$ and $h$ appropriately/accordingly

__This calculator will:__

(1.) Calculate the definite integral of a function when given the lower limit and the upper limit of the integral.

(2.) Determine the antiderivative of a function (indefinite integral of the function).

__To use the calculator, please:__

(1.) Type the function in the textbox (the bigger textbox).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed, into the small textbox of the calculator.

(5.) Type the lower and upper limits of integration in the two small textboxes of the calculator.

The first small textbox is the lower limit of integration.

The second small textbox is the upper limit of integration.

*entre means "enter", y means "and"...got to learn Spanish 😊*

(6.) Click the "CALCULAR YA..." button ("Calculate" button).

(7.) **Check to make sure that it is the correct function you typed.**

(8.) Review the answers. At least one of the answers is probably what you need.

- Using the Definite Integrals Calculator
- Type: $7$
__as__7 - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $y = 4x + 3$
__as__y = 4 * x + 3 - Type: $4x^3 - 5x^{\dfrac{1}{2}} + 7x^{-\dfrac{2}{3}}$
__as__4 * x^3 - 5 * x^(1/2) + 7 * x^(-2/3) - Type: $y = 4x^3 - 5x^2 + 4$
__as__y = 4 * x^2 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $y = |-7 - 5x|$
__as__y = |-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $y = \sec^2 x$
__as__y = sec^2 x - Type: $y = \cos hx$
__as__y = cos hx - Type: $y = \dfrac{1}{1 - x^2}$
__as__y = 1 / (1 - x^2) - Type: $y = \dfrac{-1}{\sqrt{1 - x^2}}$
__as__y = -1 / (sqrt(1 - x^2))

Integrate $wrt\;x$

Use only $x$ and $y$

$x$ is the independent vavariable.

$y$ is the dependent variable.

Use $e$ and $h$ appropriately/accordingly

__This calculator will:__

(1.) Calculate the average value of a function over an interval of values.

__To use the calculator, please:__

(1.) Type the function in the textbox (the bigger textbox).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed, into the small textbox of the calculator.

(5.) Type the lower limit and the upper limit of integration accordingly in the textboxes of the calculator.

(6.) Click the "Submit" button.

(7.) **Check to make sure that it is the correct function you typed.**

(8.) Review the answer.

- Using the Average Value Calculator
- Type: $7$
__as__7 - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $y = 4x + 3$
__as__y = 4 * x + 3 - Type: $4x^3 - 5x^{\dfrac{1}{2}} + 7x^{-\dfrac{2}{3}}$
__as__4 * x^3 - 5 * x^(1/2) + 7 * x^(-2/3) - Type: $y = 4x^3 - 5x^2 + 4$
__as__y = 4 * x^2 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $y = |-7 - 5x|$
__as__y = |-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $y = \sec^2 x$
__as__y = sec^2 x - Type: $y = \cos hx$
__as__y = cos hx - Type: $y = \dfrac{1}{1 - x^2}$
__as__y = 1 / (1 - x^2) - Type: $y = \dfrac{-1}{\sqrt{1 - x^2}}$
__as__y = -1 / (sqrt(1 - x^2))

Function:

__This calculator will:__

(1.) Determine the double integral of a function with respect to several variables.

__To use the calculator, please:__

(1.) Type the function in the textbox (the bigger textbox).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed, into the small textbox of the calculator.

(5.) Select the two variables for which you want to find the antiderivative with respect to,
as well as the order in which the integration is to be performed.

(6.) Type the lower and upper limits of integration for the first variable.

(7.) Type the lower and upper limits of integration for the second variable.

(8.) Click the "Submit" button.

(9.) **Check to make sure that it is the correct function you typed.**

(10.) Review the answers. At least one of the answers is probably what you need.

- Using the Double Integrals Calculator
- Type: $7$
__as__7 - Type: $4x + 3 - 2y$
__as__4 * x + 3 * x - 2 * y - Type: $4x^3 - 5y^{\dfrac{1}{2}} + 7x^{-\dfrac{2}{3}}$
__as__4 * x^3 - 5 * y^(1/2) + 7 * x^(-2/3) - Type: $4y^3 - 5x^2 + 4$
__as__4 * y^3 - 5 * x^2 + 4 - Type: $(-7x^3 - 2y^{-4})^{-3}$
__as__(-7 * x^3 - 2 * y^(-4))^(-3) - Type: $12e^{-3xy}$
__as__12 * e^(-3 * x * y) - Type: $\dfrac{y^2}{1 - x^2}$
__as__y^2 / (1 - x^2) - Type: $\dfrac{-x}{\sqrt{1 - y^2}}$
__as__-x / (sqrt(1 - y^2))

Example: Integrate $wrt\;y\;\;then\;\;wrt\;x$

__This calculator will:__

(1.) Calculate the double integral of a function with respect to $r$, then with respect to $\theta$

__To use the calculator, please:__

(1.) Type the function in the textbox (the bigger textbox).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed, into the small textbox of the calculator.

(5.) Type the lower and upper limits of integration for $r$.

(6.) Type the lower and upper limits of integration for $\theta$.

(7.) Click the "Submit" button.

(8.) **Check to make sure that it is the correct function you typed.**

(9.) Review the answers. At least one of the answers is probably what you need.

- Using the Double Integrals in Polar Coordinates Calculator
- Type: $7$
__as__7 - Type: $(r\sin\theta)r\;dr\;d\theta$
__as__r\sin\theta*Do not include the "other" $r$ because it is already included in the calculator* - Type: $(1 - r^2)r\;dr\;d\theta$
__as__(1 - r^2)*Do not include the "other" $r$ because it is already included in the calculator* - Type: $\pi$
__as__pi

Example: Integrate $wrt\;r\;\;then\;\;wrt\;\theta$

__This calculator will:__

(1.) Determine the triple integral of a function with respect to several variables.

__To use the calculator, please:__

(1.) Review all the directions of the previous integrals.

- Using the Triple Integrals Calculator
- Type: $7$
__as__7 - Type: $4x + 3 - 2y + 3z$
__as__4 * x + 3 * x - 2 * y + 3 * z - Type: $4x^3 - 5y^{\dfrac{1}{2}} + 7z^{-\dfrac{2}{3}}$
__as__4 * x^3 - 5 * y^(1/2) + 7 * z^(-2/3) - Type: $4y^3 - 5x^2 + 4z^4$
__as__4 * y^3 - 5 * x^2 + 4 * z^4 - Type: $(-7x^3 - 2y^{-4})^{-3}$
__as__(-7 * x^3 - 2 * y^(-4))^(-3) - Type: $12e^{-3xyz}$
__as__12 * e^(-3 * x * y * z) - Type: $\dfrac{y^2}{z^2 - x^2}$
__as__y^2 / (z^2 - x^2) - Type: $\dfrac{-x}{\sqrt{1 - y^2}}$
__as__-x / (sqrt(1 - y^2))

Example: Integrate $wrt\;z\;\;then\;\;wrt\;\;y\;\;then\;\;wrt\;x$

__This calculator will:__

(1.) Calculate the triple integral of a function with respect to $z$, then with respect to $r$, then with respect to $\theta$

__To use the calculator, please:__

(1.) Type the function in the textbox (the bigger textbox).

(2.) Type it according to the examples I listed.

(3.) Copy and paste the function you typed, into the small textbox of the calculator.

(4.) Type the lower and upper limits of integration for $z$.

(5.) Type the lower and upper limits of integration for $r$.

(6.) Type the lower and upper limits of integration for $\theta$.

(7.) Click the "Submit" button.

(8.) **Check to make sure that it is the correct function you typed.**

(9.) Review the answers. At least one of the answers is probably what you need.

- Using the Triple Integrals in Cylindrical Coordinates Calculator
- Type: $7$
__as__7 - Type: $(r\sin\theta)r\;dr\;d\theta$
__as__r\sin\theta - Type: $(1 - r^2)r\;dr\;d\theta$
__as__(1 - r^2) - Type: $z_{max} = r\cos\theta$
__as__r\cos\theta - Type: $\pi$
__as__pi

Example: Integrate $wrt\;z\;\;wrt\;r\;\;then\;\;wrt\;\theta$

__This calculator will:__

(1.) Calculate the triple integral of a function in spherical coordinates.

__To use the calculator, please:__

(1.) Review all the directions of the previous integrals.

Example:

__This calculator will:__

(1.) Calculate the triple integral of a function in spherical coordinates.

__To use the calculator, please:__

(1.) Review all the directions of the previous integrals.

Example:

Use only $x$ and $y$

$x$ is the independent vavariable.

$y$ is the dependent variable.

Use $e$ and $h$ appropriately/accordingly

__This calculator will:__

(1.) Calculate the lower limit and the upper limit of the integral.

(2.) Determine the area between two curves.

(3.) Graph the two curves and indicate the area between the curves.

__To use the calculator, please:__

(1.) Type the functions in the two textboxes (the bigger textboxes).

(2.) Type them according to the examples I listed.

(3.) Delete the "default" functions in the textboxes of the calculator.

(4.) Copy and paste the functions you typed respectively, into the small textboxes of the calculator.

(5.) Type the lower and upper limits of integration in the two small textboxes of the calculator.

The first small textbox is the lower limit of integration.

The second small textbox is the upper limit of integration.

(6.) Click the "Calculate" button.

(7.) **Check to make sure that they are the correct functions you typed.**

(8.) Review the answers. At least one of the answers is probably what you need.

- Using the Area Between Two Curves (
*not given Limits*) Calculator - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $y = 4x + 3$
__as__y = 4 * x + 3 - Type: $4x^3 - 5x^2 + 4$
__as__4 * x^3 - 5 * x^2 + 4 - Type: $y = 4x^3 - 5x^2 + 4$
__as__y = 4 * x^2 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $y = |-7 - 5x|$
__as__y = |-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $y = \sec^2 x$
__as__y = sec^2 x - Type: $y = \cos hx$
__as__y = cos hx - Type: $y = \dfrac{1}{1 - x^2}$
__as__y = 1 / (1 - x^2) - Type: $y = \dfrac{-1}{\sqrt{1 - x^2}}$
__as__y = -1 / (sqrt(1 - x^2))

First Function:

Second Function:

Use only $x$ and $y$

$x$ is the independent vavariable.

$y$ is the dependent variable.

Use $e$ and $h$ appropriately/accordingly

__This calculator will:__

(1.) Determine the area between two curves between the __given__ limits of integration.

(2.) Graph the two curves and indicate the area between the curves.

__To use the calculator, please:__

(1.) Type the functions in the two textboxes (the bigger textboxes).

(2.) Type them according to the examples I listed.

(3.) Delete the "default" functions in the textboxes of the calculator.

(4.) Copy and paste the functions you typed respectively, into the small textboxes of the calculator.

(5.) Click the "Submit" button.

(6.) **Check to make sure that they are the correct functions you typed.**

(7.) Review the answers. At least one of the answers is probably what you need.

- Using the Area Between Two Curves (
*given Limits*) Calculator - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $y = 4x + 3$
__as__y = 4 * x + 3 - Type: $4x^3 - 5x^2 + 4$
__as__4 * x^3 - 5 * x^2 + 4 - Type: $y = 4x^3 - 5x^2 + 4$
__as__y = 4 * x^2 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $y = |-7 - 5x|$
__as__y = |-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $y = \sec^2 x$
__as__y = sec^2 x - Type: $y = \cos hx$
__as__y = cos hx - Type: $y = \dfrac{1}{1 - x^2}$
__as__y = 1 / (1 - x^2) - Type: $y = \dfrac{-1}{\sqrt{1 - x^2}}$
__as__y = -1 / (sqrt(1 - x^2))

First Function:

Second Function:

__This calculator will:__

(1.) Approximate the value of a definite integral using the Trapezoidal Rule

__To use the calculator, please:__

(1.) Type the function in the first textbox (the bigger textboxes).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed into the textbox of the calculator.

(5.) Type the Number of Trapezoids, and the Lower Limit and the Upper Limit of integration, in their respective textboxes of the calculator.

(6.) Click the "Submit" button.

(7.) **Check to make sure that the function you typed is what you need.**

(8.) Review the answers. At least one of the answers is probably what you need.

- Using the Trapezoidal Rule Calculator
- Type: $7$
__as__7 - Type: $\pi$
__as__pi - Type: $\theta$
__as__theta - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $4x^3 - 5x^2 + 4$
__as__4 * x^3 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $|-7 - 5x|$
__as__|-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $\sec^2 x$
__as__sec^2 x - Type: $\cos hx$
__as__cos hx - Type: $\dfrac{1}{1 - x^2}$
__as__1 / (1 - x^2) - Type: $\dfrac{-1}{\sqrt{1 - x^2}}$
__as__-1 / (sqrt(1 - x^2))

Function:

__This calculator will:__

(1.) Approximate the value of a definite integral using the Simpson's Rule

__To use the calculator, please:__

(1.) Type the function in the first textbox (the bigger textboxes).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed into the textbox of the calculator.

(5.) Type the Lower Limit and the Upper Limit of integration, and the Interval Size, in their respective textboxes of the calculator.

(6.) Click the "Submit" button.

(7.) **Check to make sure that the function you typed is what you need.**

(8.) Review the answers. At least one of the answers is probably what you need.

- Using the Simpson's Rule Calculator
- Type: $7$
__as__7 - Type: $\pi$
__as__pi - Type: $\theta$
__as__theta - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $4x^3 - 5x^2 + 4$
__as__4 * x^3 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $|-7 - 5x|$
__as__|-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $\sec^2 x$
__as__sec^2 x - Type: $\cos hx$
__as__cos hx - Type: $\dfrac{1}{1 - x^2}$
__as__1 / (1 - x^2) - Type: $\dfrac{-1}{\sqrt{1 - x^2}}$
__as__-1 / (sqrt(1 - x^2))

Function:

__This calculator will:__

(1.) Approximate the value of a definite integral using the Simpson's Rule

__To use the calculator, please:__

(1.) Type the function in the first textbox (the bigger textboxes).

(2.) Type it according to the examples I listed.

(3.) Delete the "default" function in the textbox of the calculator.

(4.) Copy and paste the function you typed into the textbox of the calculator.

(5.) Type the Lower Limit and the Upper Limit of integration, and the Interval Size, in their respective textboxes of the calculator.

(6.) Click the "Submit" button.

(7.) **Check to make sure that the function you typed is what you need.**

(8.) Review the answers. At least one of the answers is probably what you need.

- Using the Simpson's Rule Calculator
- Type: $7$
__as__7 - Type: $\pi$
__as__pi - Type: $\theta$
__as__theta - Type: $4x + 3$
__as__4 * x + 3 * x - Type: $4x^3 - 5x^2 + 4$
__as__4 * x^3 - 5 * x^2 + 4 - Type: $(-7x^3 - 2x^{-4})^{-3}$
__as__(-7 * x^3 - 2 * x^(-4))^(-3) - Type: $|-7 - 5x|$
__as__|-7 - 5x| - Type: $12e^{-3x}$
__as__12 * e^(-3 * x) - Type: $(\ln x)^5$
__as__(log_e x)^5*Notice the underscore between log and e. Notice the space between e and x* - Type: $(\log x)^5$
__as__(log x)^5 - Type: $\sec^2 x$
__as__sec^2 x - Type: $\cos hx$
__as__cos hx - Type: $\dfrac{1}{1 - x^2}$
__as__1 / (1 - x^2) - Type: $\dfrac{-1}{\sqrt{1 - x^2}}$
__as__-1 / (sqrt(1 - x^2))

Function: