FToC and Definite Integrals

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What is the Fundamental Theorem of Calculus (FTC)?

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The Fundamental Theorem of Calculus states that if F is an antiderivative of f on an interval [a, b], then: \( \int_a^b f(x)dx = F(b) - F(a) \).

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How do you find the derivative of an integral function?

Back

To find the derivative of an integral function \( F(x) = \int_a^x f(t)dt \), use the Fundamental Theorem of Calculus: \( F'(x) = f(x) \).

Front

What is the relationship between definite integrals and area under a curve?

Back

The definite integral of a function over an interval gives the net area between the function and the x-axis over that interval.

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What does the notation \( F'(x) \) represent?

Back

The notation \( F'(x) \) represents the derivative of the function F with respect to x.

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If \( F(x) = \int_0^x \sqrt{1+t^3}dt \), what is \( F'(2) \)?

Back

Using the Fundamental Theorem of Calculus, \( F'(2) = \sqrt{1+2^3} = \sqrt{9} = 3 \).

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What is an antiderivative?

Back

An antiderivative of a function f is a function F such that \( F' = f \).

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What is the significance of the limits of integration in a definite integral?

Back

The limits of integration define the interval over which the function is being integrated, determining the bounds of the area calculated.

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How do you evaluate a definite integral?

Back

To evaluate a definite integral \( \int_a^b f(x)dx \), find an antiderivative F of f, then compute \( F(b) - F(a) \).

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What is the geometric interpretation of the definite integral?

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The geometric interpretation of the definite integral is the area under the curve of the function f(x) from x=a to x=b.

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What is the difference between indefinite and definite integrals?

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Indefinite integrals represent a family of functions (antiderivatives) without limits, while definite integrals compute a specific numerical value over an interval.