Integration — Finding Area Under a Curve
Transcript
What if you could find the area under any curve — that is the magic of integration. Let us start by drawing a simple curve and the area we want to find beneath it. The shaded region between the curve and the x-axis is exactly what integration measures.
Before integration, people tried to estimate this area using thin rectangles underneath the curve. The more rectangles you use, the closer you get to the true area under the curve. Integration is the limit of this process — infinitely many infinitely thin rectangles.
- More rectangles → more accurate area
- Integration = limit of infinite rectangles
We write integration using this special elongated S symbol, called the integral sign.
\int x^2 \, dxThe integral sign means sum, x squared is the function, and d x means a tiny strip of width. The general power rule for integration says: raise the power by one, then divide by the new power.
- dx = infinitely thin width strip
\int x^n \, dx = \frac{x^{n+1}}{n+1} + CHere n is the original power, n plus one is the new power, and C is the constant of integration. Let us work through a simple example: integrate x squared with respect to x.
- C = constant of integration (unknown shift)
\int x^2 \, dxThe power is 2, so we add 1 to get 3, giving us x cubed on top.
= \frac{x^{2+1}}{2+1} + CSimplifying, two plus one is three, so the answer is x cubed over three plus C.
= \frac{x^3}{3} + CNow let us find a definite integral — the actual area between x equals 1 and x equals 3.
\int_1^3 x^2 \, dxWe substitute the limits: plug in 3 first, then subtract the result when we plug in 1.
= \left[\frac{x^3}{3}\right]_1^3 = \frac{3^3}{3} - \frac{1^3}{3}Three cubed is 27, so 27 over 3 is 9, and 1 over 3 stays as one third.
= 9 - \frac{1}{3} = \frac{26}{3}So the exact area under x squared from 1 to 3 is twenty-six thirds, roughly 8.67 square units. A very common mistake: students forget to add C when there are no limits given.
- Area = 26/3 ≈ 8.67 sq units
\int x^2 \, dx = \frac{x^3}{3} \quad \color{red}{\times} \quad \text{Missing } C!In real life, engineers use integration to find the distance a car travels from its speed-time graph. The area under the speed-time graph — found by integration — gives the total distance travelled. Now your turn: what is the integral of x cubed with respect to x?
- Speed × Time = Distance = integral
Integration finds the area under a curve by summing infinite thin strips using the power rule.
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Quick quiz
Q1. What is ∫ x³ dx ?
Add 1 to the power (3+1=4) then divide by the new power 4, giving x⁴/4 + C.
