Calculate the area under a power function f(x) = axⁿ over a specific interval using the Fundamental Theorem of Calculus.
Function Definition
Integration Result
2.6667sq units
Area Under Curve
Antiderivative\frac{1x^{3}}{3}
Range[0, 2]
Step-by-Step Integration
#1Increase exponent: n+1 = 3.
#2F(x) = \frac{1x^{3}}{3}.
#3Evaluate: F(2) - F(0) = 2.6667 - 0.0000.
#4Final Area: 2.6667.
About this calculator
The Power Rule for Integration
To integrate a term axⁿ, we increase the exponent by 1 and divide by that new exponent. For a definite integral, we evaluate this antiderivative at the upper limit and subtract its value at the lower limit.
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Pro Tips
Power Rule: ∫ xⁿ dx = (1/(n+1))x^{n+1} (for n ≠ -1).
Natural Log: If n = -1, the integral is a ln|x|.
Geometry: The definite integral represents the signed area between the curve and the x-axis.
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Fun Facts
"The symbol for integration (∫) is an elongated 'S', standing for 'sum'."
"The Fundamental Theorem of Calculus links derivatives and integrals as inverse operations."
"Integration is used to calculate center of mass, moments of inertia, and probabilities."
