Trapezoidal Rule Calculator

Approximate the definite integral of f(x) using the trapezoidal rule. Split [a, b] into n subintervals and view the result, step size, and each sample point.

Input

Enter a function f(x), the limits a and b, and the number of subintervals n to approximate the definite integral with the trapezoidal rule.

Integrand f(x)

e.g. sin(x), x^2, exp(-x^2). Supports pi, e, and implicit multiplication such as 2x.

Lower limit a

Upper limit b

Subintervals n

Result

Approximation of ∫ from 0 to 3.1416 of f(x) dx

1.9998355039

Subintervals n

100

Step size h

0.0314159265

Sample points

101

Sample points

Rows are thinned out when there are many points. Endpoints have weight 1 and interior points have weight 2.

k	x	f(x)	Weight
0	0	0	1
1	0.0314159265	0.0314107591	2
2	0.0628318531	0.0627905195	2
3	0.0942477796	0.0941083133	2
4	0.1256637061	0.1253332336	2
5	0.1570796327	0.156434465	2
6	0.1884955592	0.1873813146	2
7	0.2199114858	0.2181432414	2
8	0.2513274123	0.2486898872	2
9	0.2827433388	0.278991106	2
10	0.3141592654	0.3090169944	2
11	0.3455751919	0.3387379202	2
12	0.3769911184	0.3681245527	2
13	0.408407045	0.3971478906	2
14	0.4398229715	0.4257792916	2
15	0.471238898	0.4539904997	2
16	0.5026548246	0.4817536741	2
17	0.5340707511	0.5090414158	2
18	0.5654866776	0.535826795	2
19	0.5969026042	0.5620833779	2
20	0.6283185307	0.5877852523	2
21	0.6597344573	0.6129070537	2
22	0.6911503838	0.6374239897	2
23	0.7225663103	0.6613118653	2
24	0.7539822369	0.6845471059	2
25	0.7853981634	0.7071067812	2
26	0.8168140899	0.7289686274	2
27	0.8482300165	0.7501110696	2
28	0.879645943	0.7705132428	2
29	0.9110618695	0.7901550124	2
30	0.9424777961	0.8090169944	2
31	0.9738937226	0.8270805743	2
32	1.0053096491	0.8443279255	2
33	1.0367255757	0.860742027	2
34	1.0681415022	0.87630668	2
35	1.0995574288	0.8910065242	2
36	1.1309733553	0.9048270525	2
37	1.1623892818	0.9177546257	2
38	1.1938052084	0.9297764859	2
39	1.2252211349	0.940880769	2
40	1.2566370614	0.9510565163	2
41	1.288052988	0.9602936857	2
42	1.3194689145	0.9685831611	2
43	1.350884841	0.9759167619	2
44	1.3823007676	0.9822872507	2
45	1.4137166941	0.9876883406	2
46	1.4451326207	0.9921147013	2
47	1.4765485472	0.9955619646	2
48	1.5079644737	0.9980267284	2
49	1.5393804003	0.9995065604	2
50	1.5707963268	1	2
51	1.6022122533	0.9995065604	2
52	1.6336281799	0.9980267284	2
53	1.6650441064	0.9955619646	2
54	1.6964600329	0.9921147013	2
55	1.7278759595	0.9876883406	2
56	1.759291886	0.9822872507	2
57	1.7907078125	0.9759167619	2
58	1.8221237391	0.9685831611	2
59	1.8535396656	0.9602936857	2
60	1.8849555922	0.9510565163	2
61	1.9163715187	0.940880769	2
62	1.9477874452	0.9297764859	2
63	1.9792033718	0.9177546257	2
64	2.0106192983	0.9048270525	2
65	2.0420352248	0.8910065242	2
66	2.0734511514	0.87630668	2
67	2.1048670779	0.860742027	2
68	2.1362830044	0.8443279255	2
69	2.167698931	0.8270805743	2
70	2.1991148575	0.8090169944	2
71	2.230530784	0.7901550124	2
72	2.2619467106	0.7705132428	2
73	2.2933626371	0.7501110696	2
74	2.3247785637	0.7289686274	2
75	2.3561944902	0.7071067812	2
76	2.3876104167	0.6845471059	2
77	2.4190263433	0.6613118653	2
78	2.4504422698	0.6374239897	2
79	2.4818581963	0.6129070537	2
80	2.5132741229	0.5877852523	2
81	2.5446900494	0.5620833779	2
82	2.5761059759	0.535826795	2
83	2.6075219025	0.5090414158	2
84	2.638937829	0.4817536741	2
85	2.6703537556	0.4539904997	2
86	2.7017696821	0.4257792916	2
87	2.7331856086	0.3971478906	2
88	2.7646015352	0.3681245527	2
89	2.7960174617	0.3387379202	2
90	2.8274333882	0.3090169944	2
91	2.8588493148	0.278991106	2
92	2.8902652413	0.2486898872	2
93	2.9216811678	0.2181432414	2
94	2.9530970944	0.1873813146	2
95	2.9845130209	0.156434465	2
96	3.0159289474	0.1253332336	2
97	3.047344874	0.0941083133	2
98	3.0787608005	0.0627905195	2
99	3.1101767271	0.0314107591	2
100	3.1415926536	0	1

How it works

The composite trapezoidal rule divides the interval [a, b] into n equal subintervals and approximates each piece as a trapezoid, then sums the areas.
The step size is h = (b - a) / n and the sample points are x_k = a + k·h for k = 0, 1, …, n.
The approximation is h × ( f(x0)/2 + f(x1) + … + f(x_{n-1}) + f(xn)/2 ). Endpoints carry weight 1 and interior points carry weight 2.
Increasing n reduces the error and converges toward the true value; for a smooth f the error decreases roughly in proportion to h².
Expressions are evaluated by a built-in safe parser supporting + - * / ^, parentheses, implicit multiplication, the variable x, the constants pi and e, and functions such as sin, cos, tan, exp, log, and sqrt.

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