Simpson's (Parabolic) Rule Calculator

Online calculator for approximating the definite integral using the Simpson's (Parabolic) rule, with steps shown.

Show Instructions
  • In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`.
  • In general, you can skip parentheses, but be very careful: e^3x is `e^3x`, and e^(3x) is `e^(3x)`.
  • Also, be careful when you write fractions: 1/x^2 ln(x) is `1/x^2 ln(x)`, and 1/(x^2 ln(x)) is `1/(x^2 ln(x))`.
  • If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
  • Sometimes I see expressions like tan^2xsec^3x: this will be parsed as `tan^(2*3)(x sec(x))`. To get `tan^2(x)sec^3(x)`, use parentheses: tan^2(x)sec^3(x).
  • Similarly, tanxsec^3x will be parsed as `tan(xsec^3(x))`. To get `tan(x)sec^3(x)`, use parentheses: tan(x)sec^3(x).
  • From the table below, you can notice that sech is not supported, but you can still enter it using the identity `sech(x)=1/cosh(x)`.
  • If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
  • All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
TypeGet
Constants
ee
pi`pi`
ii (imaginary unit)
Operations
a+ba+b
a-ba-b
a*b`a*b`
a^b, a**b`a^b`
sqrt(x), x^(1/2)`sqrt(x)`
cbrt(x), x^(1/3)`root(3)(x)`
root(x,n), x^(1/n)`root(n)(x)`
x^(a/b)`x^(a/b)`
abs(x)`|x|`
Functions
e^x`e^x`
ln(x), log(x)ln(x)
ln(x)/ln(a)`log_a(x)`
Trigonometric Functions
sin(x)sin(x)
cos(x)cos(x)
tan(x)tan(x), tg(x)
cot(x)cot(x), ctg(x)
sec(x)sec(x)
csc(x)csc(x), cosec(x)
Inverse Trigonometric Functions
asin(x), arcsin(x), sin^-1(x)asin(x)
acos(x), arccos(x), cos^-1(x)acos(x)
atan(x), arctan(x), tan^-1(x)atan(x)
acot(x), arccot(x), cot^-1(x)acot(x)
asec(x), arcsec(x), sec^-1(x)asec(x)
acsc(x), arccsc(x), csc^-1(x)acsc(x)
Hyperbolic Functions
sinh(x)sinh(x)
cosh(x)cosh(x)
tanh(x)tanh(x)
coth(x)coth(x)
1/cosh(x)sech(x)
1/sinh(x)csch(x)
Inverse Hyperbolic Functions
asinh(x), arcsinh(x), sinh^-1(x)asinh(x)
acosh(x), arccosh(x), cosh^-1(x)acosh(x)
atanh(x), arctanh(x), tanh^-1(x)atanh(x)
acoth(x), arccoth(x), cot^-1(x)acoth(x)
acosh(1/x)asech(x)
asinh(1/x)acsch(x)

Enter a function: `f=`

Enter a lower limit: `a=`

Enter an upper limit: `b=`

Enter the number of rectangles: `n=`

Write all suggestions in comments below.

Solution

Your input: approximate the integral $$$\int_{0}^{1}\frac{1}{\sqrt[3]{x^{5} + 7}}\ dx$$$ using $$$n=4$$$ rectangles.

The Simpson's rule states that $$$\int_{a}^{b}f(x)dx\approx\frac{\Delta{x}}{3}\left(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+...+4f(x_{n-2})+2f(x_{n-1})+f(x_n)\right)$$$, where $$$\Delta{x}=\frac{b-a}{n}$$$.

We have that $$$a=0$$$, $$$b=1$$$, $$$n=4$$$.

Therefore, $$$\Delta{x}=\frac{1-0}{4}=\frac{1}{4}$$$.

Divide the interval $$$\left[0,1\right]$$$ into $$$n=4$$$ subintervals of length $$$\Delta{x}=\frac{1}{4}$$$, with the following endpoints: $$$a=0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1=b$$$.

Now, we just evaluate the function at those endpoints:

$$$f\left(x_{0}\right)=f(a)=f\left(0\right)=\frac{7^{\frac{2}{3}}}{7}=0.52275795857471$$$

$$$4f\left(x_{1}\right)=4f\left(\frac{1}{4}\right)=\frac{32}{7169} \sqrt[3]{2} \left(7169\right)^{\frac{2}{3}}=2.09093460413808$$$

$$$2f\left(x_{2}\right)=2f\left(\frac{1}{2}\right)=\frac{4}{15} \sqrt[3]{15} \cdot 2^{\frac{2}{3}}=1.0439647043117$$$

$$$4f\left(x_{3}\right)=4f\left(\frac{3}{4}\right)=\frac{32}{7411} \sqrt[3]{2} \left(7411\right)^{\frac{2}{3}}=2.06792304223835$$$

$$$f\left(x_{4}\right)=f(b)=f\left(1\right)=\frac{1}{2}=0.5$$$

Finally, just sum up the above values and multiply by $$$\frac{\Delta{x}}{3}=\frac{1}{12}$$$: $$$\frac{1}{12}(0.52275795857471+2.09093460413808+1.0439647043117+2.06792304223835+0.5)=0.518798359105237$$$

Answer: 0.518798359105237.