Understanding Integration Applications
Area Under a Curve
Use definite integrals to find areas between curves and the x-axis. This method works for both positive and negative function intervals.
Area = ∫ab |f(x)| dx
Accumulated Quantities
Integrals can calculate total accumulation over time intervals, such as total distance from a velocity function.
Total = ∫t₁t₂ v(t) dt
Example: Area Between Two Curves
Find the area between f(x) = x² and g(x) = 2x from x = 0 to 2
Solution: ∫₀² (2x - x²) dx = [x² - (x³)/3] from 0 to 2 = 4 - 8/3 = 4/3
Interactive Visualization
Integration Explorer
Adjust the function parameters to see how the area under the curve changes dynamically.
Open Visualization ToolTry These Problems
Problem 1
Find the area under y = 2x + 3 between x = 1 and x = 4
Problem 2
Calculate ∫₋₁¹ |x| dx