# Trapezoidal Rule Calculator

The calculator will approximate the integral using the Trapezoidal Rule, with steps shown.

• 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:
 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) 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 $$\int_{0}^{1}\sqrt{\sin^{3}{\left (x \right )} + 1}\ dx$$$using $$n=5$$$ rectangles.

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

We have that $$a=0$$$, $$b=1$$$, $$n=5$$$. Therefore, $$\Delta{x}=\frac{1-0}{5}=\frac{1}{5}$$$.

Divide the interval $$\left[0,1\right]$$$into $$n=5$$$ subintervals of length $$\Delta{x}=\frac{1}{5}$$$, with the following endpoints: $$a=0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1=b$$$.

Now, we just evaluate the function at those endpoints:

$$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$$$$$2f\left(x_{1}\right)=2f\left(\frac{1}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{1}{5} \right )} + 1}=2.00782606791279$$$

$$2f\left(x_{2}\right)=2f\left(\frac{2}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{2}{5} \right )} + 1}=2.05820697233265$$$$$2f\left(x_{3}\right)=2f\left(\frac{3}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{3}{5} \right )} + 1}=2.17257446116512$$$

$$2f\left(x_{4}\right)=2f\left(\frac{4}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{4}{5} \right )} + 1}=2.34021475342487$$$$$f\left(x_{5}\right)=f(b)=f\left(1\right)=\sqrt{\sin^{3}{\left (1 \right )} + 1}=1.26325897447473$$$

Finally, just sum up the above values and multiply by $$\frac{\Delta{x}}{2}=\frac{1}{10}$$$: $$\frac{1}{10}(1+2.00782606791279+2.05820697233265+2.17257446116512+2.34021475342487+1.26325897447473)=1.08420812293102$$$