Integrals
Area under a curve
Build area from thin vertical strips. Separate geometric area from signed accumulation when the graph falls below the axis.
1. A growing region
1 / 3A(b)=\int_0^b x^2\,dx=\frac{b^3}{3} Under y=x² from 0 to b≥0, the area is b³/3. Increasing the endpoint adds positive strips. The area depends on the whole interval, not only the final height.
- b
- 1
- f(b)
- 1
- A(b)
- 0.333333
Shading shows the region between the bounds. With bounds in increasing order, regions below the axis contribute negatively; reversing the bounds reverses the integral’s sign. Geometric area stays nonnegative.
2. Rectangle sums
2 / 3R_n=\sum_{k=1}^n\left(\frac{2k}{n}\right)^2\frac2n Approximate the area under x² on [0,2] using n right-endpoint rectangles. Since x² increases there, these rectangles overestimate the integral 8/3.
- n
- 4
- Rₙ
- 3.75
- ∫
- 2.666667
Rectangles use right endpoints. Their signed sum approximates the integral; increasing the number of rectangles reduces the error in this example.
3. Signed area
3 / 3\int_{-1}^b x\,dx=\frac{b^2-1}{2} For y=x on [−1,1], the negative triangle and positive triangle cancel in the integral. Their geometric areas add instead. Move the upper endpoint to compare the signed accumulation.
- ∫
- 0
- b
- 1
Shading shows the region between the bounds. With bounds in increasing order, regions below the axis contribute negatively; reversing the bounds reverses the integral’s sign. Geometric area stays nonnegative.