Key takeaways
- For powers of x: ∫ from a to b of xⁿ dx = (bⁿ⁺¹ − aⁿ⁺¹) ÷ (n + 1).
- Add 1 to the power, divide by the new power, then do top limit minus bottom limit.
- Area below the x-axis counts as negative.
- The rule fails for n = −1 (use ln) and for negative powers on an interval that includes 0.
In this guide
What is a definite integral?
Picture the graph of y = x² between x = 0 and x = 3. The definite integral is the area between the curve and the x-axis in that stretch. You can think of it as slicing the region into very thin rectangles and adding them all up.
Integration is the reverse of differentiation. A derivative turns a total into a rate (distance into speed). An integral turns a rate back into a total (speed into distance).
Why do definite integrals matter?
Many real totals build up from a rate that keeps changing. A car’s speed changes every second, so you can’t just multiply speed × time. Water flowing into a tank, a force that grows as a spring stretches, electricity used through a day: all of these are areas under a curve.
The ancient Greek mathematician Archimedes found the area of a region cut from a parabola by filling it with more and more triangles. With the power rule, the same result takes one line.
The formula
| Symbol | What it means | Example |
|---|---|---|
| ∫ | The integral sign, a stretched “S” for “sum” | ∫₀³ |
| n | The power of x (any number except −1) | 2 |
| a | Lower limit: where the area starts | 0 |
| b | Upper limit: where the area ends | 3 |
| dx | Shows that x is the variable you add along | dx |
The rule in words: add 1 to the power, then divide by the new power. Then put in b, put in a, and subtract.
How to solve a definite integral step by step
- 1Raise the power by 1. xⁿ becomes xⁿ⁺¹.
- 2Divide by the new power. You now have
xⁿ⁺¹ ÷ (n + 1). - 3Put in the upper limit b. Work out the value.
- 4Put in the lower limit a. Work out that value too.
- 5Subtract. Answer = (value at b) − (value at a).
Now try it yourself. Enter the power and the two limits in the calculator below to see each step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using n = −1. x⁻¹ = 1/x would mean dividing by zero. Its integral is ln|x|, so the calculator shows an error for n = −1.
- Crossing a “hole” in the graph. For negative powers like x⁻², the curve shoots up to infinity at x = 0. If your interval includes 0, the calculator tells you the integral is improper and the power rule can’t be used directly.
- Expecting area to always be positive. Area below the x-axis counts as negative:
∫ from −1 to 0 of x dx = −0.5. - Entering a ≥ b. This calculator needs the lower limit to be smaller than the upper limit.
Tips and tricks
- No + C needed. The constant cancels when you subtract, so definite integrals don’t need it.
- Estimate first. Sketch the curve and guess the area as a rectangle or triangle. Your answer should be in the same ballpark.
- See the rectangles. To watch the area being built from strips, try the Riemann sum calculator or the trapezoidal rule calculator.
Where you'll use it in real life
- Motion: total distance travelled from a speed formula like
v = t². - Engineering: work done by a force that changes as something moves.
- Design: the area of a curved garden bed, window or logo.
- Statistics: probabilities are areas under curves, too.
Quick summary
To find ∫ₐᵇ xⁿ dx, add 1 to the power, divide by the new power, then subtract the value at a from the value at b. Avoid n = −1 and negative powers across 0. Use the calculator above for the area with every step shown (it rounds to 2 decimal places), and use the power rule for derivatives to go the other way.
Worked example
A designer is shaping a curved garden bed along y = x², with x from 0 to 3 metres. How many square metres of soil area lie under the curve? The power is n = 2, the lower limit is a = 0 and the upper limit is b = 3.
- 01∫[0 to 3] x^2 dx = [x^3/3]|[0 to 3]
- 02= (3^3/3) - (0^3/3)
- 03= 9 - 0
- 04= 9
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the difference between a definite and an indefinite integral?
A definite integral has limits (a and b) and gives a number, like an area. An indefinite integral has no limits and gives a formula plus a constant C.
Why don’t I add + C in a definite integral?
The constant C appears in both the upper and lower values, so it cancels when you subtract. The answer is the same whatever C is.
Can a definite integral be negative?
Yes. If the curve is below the x-axis, that part of the area counts as negative. For example, ∫ from −1 to 0 of x dx is −0.5.
What is the integral of x² from 1 to 2?
Raise the power and divide: x³/3. At 2 it is 8/3 and at 1 it is 1/3, so the answer is 7/3, about 2.33.
