From Area Under a Curve to the Fundamental Theorem
A definite integral ∫ab f(x) dx is the signed area between the curve and the x-axis from a to b. Area above the axis counts positive, area below counts negative, so ∫02π sin x dx is 0 even though the shape is not empty.
The Fundamental Theorem of Calculus is why this is easy: find any antiderivative F, then the answer is F(b) − F(a). Newton and Leibniz joined two ideas that had looked unrelated, slopes and areas.
∫₀¹ x² dx
- Antiderivative: F(x) = x³/3.
- F(1) − F(0) = 1/3 − 0.
Answer: 1/3 ≈ 0.3333
∫₀^π sin x dx
- F(x) = −cos x.
- F(π) − F(0) = (−cos π) − (−cos 0) = 1 + 1.
Answer: 2
Riemann Sums: How the Integral Is Built
Cut [a, b] into n strips and add up their areas. The shape of the top of each strip decides the method. Below is ∫01 x² dx with n = 4, where the exact answer is 0.33333.
| Method | Value | Error |
|---|---|---|
| Left | 0.21875 | −0.1146 |
| Right | 0.46875 | +0.1354 |
| Midpoint | 0.328125 | −0.0052 |
| Trapezoid | 0.34375 | +0.0104 |
| Simpson | 0.33333 | 0 |
Simpson is exact here because it fits parabolas and the integrand is a parabola. Left and right sums are the worst of the five, which is why serious numerical work never uses them alone.
Area Between Curves and Average Value
Area enclosed by y = x and y = x² on [0, 1]
- On [0, 1] the line is above the parabola, so integrate top minus bottom: ∫(x − x²) dx.
- F(x) = x²/2 − x³/3.
- F(1) − F(0) = 1/2 − 1/3.
Answer: 1/6 ≈ 0.1667
Average of f(x) = sin x on [0, π]
- Average = (1 ÷ (b − a)) ∫f dx.
- Integral = 2 and b − a = π.
Answer: 2/π ≈ 0.6366
A tank filling at r(t) = 3t² litres per minute holds ∫04 3t² dt = 4³ = 64 litres after four minutes. Rate in, quantity out: that is nearly every application.
Improper Integrals
An integral is improper when a limit is infinite or the function blows up inside the interval. Replace the bad endpoint with a variable and take a limit.
| Integral | Result | Why |
|---|---|---|
| ∫₁^∞ 1/x² dx | 1 | Converges: the tail shrinks fast enough |
| ∫₁^∞ 1/x dx | diverges | Grows like ln b without bound |
| ∫₀¹ 1/√x dx | 2 | Converges despite the spike at 0 |
| ∫₋∞^∞ e^(−x²) dx | √π ≈ 1.77245 | Gaussian integral, no elementary antiderivative |
The p-integral rule is worth remembering: ∫1∞ 1/xp dx converges exactly when p > 1.
What We Think Is Overrated, Underrated and Wrong
Definite Integrals People Search For
| Search | Answer | Method |
|---|---|---|
| ∫₀^π sin x dx | 2 | area of one arch |
| ∫₀¹ x² dx | 1/3 | power rule |
| ∫₋₁¹ x³ dx | 0 | odd function symmetry |
| ∫₀¹ eˣ dx | e − 1 ≈ 1.71828 | exponential |
| ∫₁^∞ 1/x² dx | 1 | improper, converges |
| ∫₁^∞ 1/x dx | diverges | improper, p = 1 |
Original Data: Riemann Sums Closing In
| Rectangles n | Right-endpoint sum of x² on [0, 1] | Error vs 1/3 |
|---|---|---|
| 10 | 0.38500 | +0.0517 |
| 100 | 0.33835 | +0.0050 |
| True value | 0.33333 | 0 |
The error shrinks roughly like 1 ÷ n, which is why the exact antiderivative method beats counting rectangles. The sums come from (1 ÷ n³) × Σk².
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: Riemann sums as a calculating tool. They explain the idea, and then they should be retired.
Underrated: checking the sign of the answer first. A function above the axis must give a positive integral.
Plain wrong: treating the integral as area. ∫₀^(2π) sin x dx = 0, while the actual area is 4.
Definite Integral Mistakes That Cost Marks
- Subtracting in the wrong order. It is F(b) − F(a).
- Using area = integral when the curve crosses the axis.
- Forgetting to change the limits after a u-substitution, or to convert back.
- Integrating straight through a vertical asymptote such as 1/x on [−1, 1].
- Using degrees in a trig integral. Calculus assumes radians.
Why Students Get Stuck on Definite Integrals, and What a Good Calculator Should Do
Searching for help with definite integrals usually starts late at night with one problem that will not work out. The antiderivative is found, then the bounds are plugged in the wrong way round. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to connect the number to an area and evaluate the bounds correctly, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Subtracting F(a) − F(b) instead of F(b) − F(a)Integrate x^2 + 1 from 0 to 2↗
- Forgetting to change the bounds after a substitutionIntegrate sin(x) from 0 to pi↗
- Treating area below the axis as positive without checkingIntegrate 1/x from 1 to e↗
- Not knowing when an improper integral convergesIntegrate x*e^(-x) from 0 to 1↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the antiderivative and the evaluation shown separately
- Wants a picture of the area under the curve
- Wants a numerical check of the result
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Integrate x^2 + 1 from 0 to 2↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.