# Riemann Sum Calculator for a Function

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

Related calculator: Riemann Sum Calculator for a Table

- 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.

Type | Get |

Constants | |

e | e |

pi | `pi` |

i | i (imaginary unit) |

Operations | |

a+b | a+b |

a-b | a-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)` |

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) |

## Your Input

**Approximate the integral with using the left Riemann sum.**

## Solution

The left Riemann sum (also known as the left endpoint approximation) uses the left endpoints of a subinterval:

where .

We have that , , .

Therefore, .

Divide the interval into subintervals of the length with the following endpoints: , , , , .

Now, just evaluate the function at the left endpoints of the subintervals.

Finally, just sum up the above values and multiply by :