Table of Contents
Definition
Integration is used to find areas, volumes, central points and often is used to find the area underneath the graph of a function.
Common Functions of Integration
| Function | Integral | |
| Constant | ∫ a dx | ax + c |
| Variable | ∫ x dx |  |
| Square | ∫ x2 dx |  |
| Reciprocal | ∫ (1/x) dx | ln|x| + c |
| Exponential | ∫ ex dx | ex + c |
| ∫ ax dx |  | |
| ∫ ln(x) dx | x ln(x) − x + c | |
| Trigonometry (x in radians) | ∫ cos(x) dx | sin(x) + c |
| ∫ sin(x) dx | -cos(x) + c | |
| ∫ sec2(x) dx | tan(x) + c |
Rules of Integration
| Rules | Function | Integral |
| Multiplication by constant | ∫ cf(x) dx | c∫f(x) dx |
| Power Rule (n≠-1) | ∫ xn dx |  |
| Sum Rule | ∫ (f + g) dx | ∫f dx + ∫g dx |
| Difference Rule | ∫ (f – g) dx | ∫f dx – ∫g dx |
| Rules | Function | Application | Integral |
| Multiplication by constant | ∫ 5 cos(x) dx | 5 ∫ cos(x) dx | 5 sin(x) + c |
| Power Rule (n ≠ -1) | ∫ x2 dx |  |  |
| Sum Rule | ∫ (cos(x) + x2) dx | ∫ cos(x) dx + ∫ x2 dx |  |
| Difference Rule | ∫ (x3 – sin(x)) dx | ∫ x3 dx – ∫ sin(x) dx |  |
Examples
7.Find the area underneath the curve y = x2 + 3 from x = -1 to x = 1.
Answer:
8. Determine the volume of the solid of the revolution generated when the region is bounded by f(x) = -2×2 + 8 and the x -axis revolved about x-axis.
Answer:
Determine intersect point
-2x2 + 8 = 0
2x2 = 8
x2 = 4
x = 2 or x = -2
