Midpoint Rule Calculator

Online calculator for approximating definite integral using the Midpoint (Mid ordinate) 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 $$$\int_{1}^{3}\sqrt{\sin^{4}{\left (x \right )} + 7}\ dx$$$ using $$$n=4$$$ rectangles.

The Midpoint Sum (also Midpoint Approximation) uses midpoints of subinterval: $$$\int_{a}^{b}f(x)dx\approx\Delta{x}\left(f\left(\frac{x_0+x_1}{2}\right)+f\left(\frac{x_1+x_2}{2}\right)+f\left(\frac{x_2+x_3}{2}\right)+...+f\left(\frac{x_{n-2}+x_{n-1}}{2}\right)+f\left(\frac{x_{n-1}+x_{n}}{2}\right)\right)$$$, where $$$\Delta{x}=\frac{b-a}{n}$$$.

We have that $$$a=1$$$, $$$b=3$$$, $$$n=4$$$.

Therefore, $$$\Delta{x}=\frac{3-1}{4}=\frac{1}{2}$$$.

Divide interval $$$\left[1,3\right]$$$ into $$$n=4$$$ subintervals of length $$$\Delta{x}=\frac{1}{2}$$$ with the following endpoints: $$$a=1, \frac{3}{2}, 2, \frac{5}{2}, 3=b$$$.

Now, we just evaluate the function at those endpoints:

$$$f\left(\frac{x_{0}+x_{1}}{2}\right)=f\left(\frac{\left(1\right)+\left(\frac{3}{2}\right)}{2}\right)=f\left(\frac{5}{4}\right)=\sqrt{\sin^{4}{\left (\frac{5}{4} \right )} + 7}=2.79482192294185$$$

$$$f\left(\frac{x_{1}+x_{2}}{2}\right)=f\left(\frac{\left(\frac{3}{2}\right)+\left(2\right)}{2}\right)=f\left(\frac{7}{4}\right)=\sqrt{\sin^{4}{\left (\frac{7}{4} \right )} + 7}=2.81735090562718$$$

$$$f\left(\frac{x_{2}+x_{3}}{2}\right)=f\left(\frac{\left(2\right)+\left(\frac{5}{2}\right)}{2}\right)=f\left(\frac{9}{4}\right)=\sqrt{\sin^{4}{\left (\frac{9}{4} \right )} + 7}=2.71413091375118$$$

$$$f\left(\frac{x_{3}+x_{4}}{2}\right)=f\left(\frac{\left(\frac{5}{2}\right)+\left(3\right)}{2}\right)=f\left(\frac{11}{4}\right)=\sqrt{\sin^{4}{\left (\frac{11}{4} \right )} + 7}=2.64975816351283$$$

Finally, just sum up the above values and multiply by $$$\Delta{x}=\frac{1}{2}$$$: $$$\frac{1}{2}(2.79482192294185+2.81735090562718+2.71413091375118+2.64975816351283)=5.48803095291652$$$

Answer: 5.48803095291652.