IntegralsMedium
Evaluate ∫₀¹ x² dx
Use an antiderivative and the Fundamental Theorem of Calculus to evaluate a definite integral.
Method: Fundamental Theorem of Calculus
Step-by-step solution
- 1
Find an antiderivative
Reverse the power rule.
F(x)= rac{x^3}{3} - 2
Evaluate the endpoints
Compute F(1)=1/3 and F(0)=0.
- 3
Subtract
Use F(1)−F(0).
rac13-0= rac13
Final answer
Because x² is nonnegative on [0,1], the signed integral also equals the geometric area under the curve.
Common mistake
A definite integral does not need +C because the constant cancels in the endpoint subtraction.
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