Calculus / Unit 7: Antiderivatives and the Integral
7.5 The Definite Integral and Area
Notes
The definite integral is the limit of Riemann sums as $n \to \infty$: $\displaystyle \int_a^b f(x)\,dx$.
- It gives net signed area: area above the $x$-axis counts positive and area below counts negative.
- For simple shapes, use geometry: triangles, rectangles, semicircles.
- Example: $\displaystyle \int_0^4 x\,dx$ is a triangle with base 4 and height 4, so the area is $\displaystyle \frac{1}{2}(4)(4) = 8$.
Practice
Find $\displaystyle \int_0^4 x\,dx$ using geometry.
The region is a triangle with area $\displaystyle \frac{1}{2}(4)(4) = 8$.
Find $\displaystyle \int_{-3}^{3} \sqrt{9 - x^2}\,dx$.
The graph is the upper half of a circle of radius 3, with area $\displaystyle \frac{1}{2}\pi(3)^2$.
Find $\displaystyle \int_0^6 (x - 3)\,dx$ using geometry.
A triangle of area 4.5 lies below the axis on $[0, 3]$ and an equal one lies above on $[3, 6]$. They cancel.
Ready for more? Take the Unit 7 test.