Riemann Sum Calculator

Calculator will approximate definite integral using Riemann sum and sample points of your choice: left endpoints, right endpoints, midpoints, trapezoids.

Show Instructions
  • In general, you can skip 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 multiplication sign, type at least 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 table below you can notice, that sech is not supported, but you can still enter it using 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 welcomed. Leave them in comments
The following table contains 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 function: `f=`

Enter lower limit: `a=`

Enter upper limit: `b=`

Enter number of rectangles: `n=`

Choose type:

Write all suggestions in comments below.

Solution

Your input: find Riemann Sum for $$$\int_{0}^{2}\sqrt[3]{x^{4} + 1}\ dx$$$ with $$$n=4$$$ rectangles, using left endpoints.

Left Riemann Sum (also Left Endpoint Approximation) uses left endpoints of subinterval: $$$\int_{a}^{b}f(x)dx\approx\Delta{x}\left(f(x_0)+f(x_1)+2f(x_2)+...+f(x_{n-2})+f(x_{n-1})\right)$$$, where $$$\Delta{x}=\frac{b-a}{n}$$$.

We have that $$$a=0$$$, $$$b=2$$$, $$$n=4$$$.

Therefore, $$$\Delta{x}=\frac{2-0}{4}=\frac{1}{2}$$$.

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

Now, we just evaluate function at left endpoints:

$$$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$$$

$$$f\left(x_{1}\right)=f\left(\frac{1}{2}\right)=\frac{2^{\frac{2}{3}}}{4} \sqrt[3]{17}=1.02041377547934$$$

$$$f\left(x_{2}\right)=f\left(1\right)=\sqrt[3]{2}=1.25992104989487$$$

$$$f\left(x_{3}\right)=f\left(\frac{3}{2}\right)=\frac{2^{\frac{2}{3}} \sqrt[3]{97}}{4}=1.82340825744217$$$

Finally, just sum up above values and multiply by $$$\Delta{x}=\frac{1}{2}$$$: $$$\frac{1}{2}(1+1.02041377547934+1.25992104989487+1.82340825744217)=2.55187154140819$$$

Answer: 2.55187154140819.