Definite integrals and substitution (HSC SSCE Mathematics Extension 1): Flashcards
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What does a definite integral measure?
What does a definite integral measure?
It measures the area under a curve between two points.
What is the purpose of substitution in integrals?
What is the purpose of substitution in integrals?
It simplifies complex integrals by changing variables.
How do you calculate a definite integral?
How do you calculate a definite integral?
Evaluate the antiderivative at the limits and subtract.
What happens when you reverse limits of integration?
What happens when you reverse limits of integration?
The sign of the integral changes.
What is a common substitution for sin²(x)?
What is a common substitution for sin²(x)?
Use sin²(x) = (1 - cos(2x))/2.
How do you change limits when substituting?
How do you change limits when substituting?
Convert x limits to u limits using the substitution.
What happens when you reverse the limits of integration?
What happens when you reverse the limits of integration?
The integral changes sign, becoming negative.
What is the integral of 1/sqrt(1-x²)?
What is the integral of 1/sqrt(1-x²)?
It equals arcsin(x) + C after substitution.
Why are definite integrals important?
Why are definite integrals important?
They are crucial for calculating areas and volumes.
What does the integral ∫(3x + 1) dx from 2 to 5 equal?
What does the integral ∫(3x + 1) dx from 2 to 5 equal?
Evaluate to find the area under the curve.