Integrals
Integration as reverse differentiation, and as area under a curve.
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Integration adds up tiny slices to find the area under a curve.
Look: The yellow area under y = x² from 0 to 3 is 9.
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1∫ x⁴ dx
2∫ from 1 to 2 of 2x dx
3∫ cos x dx
! Careful!
✗∫x² = 2x.
✓That is differentiating. Integrating gives x³/3.
✗Forgetting + C.
✓Without limits, add + C.
See the whole lesson on one page
Integration adds up tiny slices to find the area under a curve.
Integration is the opposite of differentiation. Since x³ changes at rate 3x², the integral of 3x² is x³. For an area from a to b, work out the answer at b minus the answer at a.
Think of it like thisCut a loaf into very thin slices and add up their sizes: you get the whole loaf. Integration does that with thin strips of area.
- ∫
- Integral: "add up".
Solved together
Find the area under y = x² from x = 0 to x = 3.
Raise the power by 1, divide by the new power.
x² → x³/3Value at 3 minus value at 0.
27/3 − 0 = 9
AnswerArea = 9.
For older learners and grown-ups: a little more
Area between curves = ∫(top − bottom). Between y = x and y = x² (from 0 to 1): ∫(x − x²) dx = ½ − ⅓ = 1/6.
Now practise it
Kanakku makes new questions on integrals every time. It shows the working and gives a hint when you are stuck.