Definite Integrals (part 4) | Summary and Q&A

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October 22, 2007
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Definite Integrals (part 4)

TL;DR

The fundamental theorem of calculus states that the definite integral of a function can be calculated by evaluating its antiderivative at the upper and lower limits. This allows for the calculation of areas between curves.

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Questions & Answers

Q: What does the fundamental theorem of calculus state?

The fundamental theorem of calculus states that if the derivative of a function is equal to the function itself, then the definite integral of the function can be calculated by evaluating its antiderivative at the upper and lower limits.

Q: How can the definite integral be used to calculate the area between a curve and the x-axis?

By finding the antiderivative of the function and evaluating it at the upper and lower limits, the definite integral gives the area between the curve and the x-axis.

Q: How can the area between two curves be calculated?

To calculate the area between two curves, the areas under each individual curve are calculated separately using the definite integral, and then the result of the second curve is subtracted from the result of the first curve.

Q: Why does the constant in the antiderivative not affect the final result?

When subtracting the antiderivative values at the upper and lower limits, the constants cancel out, so the exact value of the constant does not matter in calculating the definite integral.

Summary & Key Takeaways

  • The fundamental theorem of calculus states that if a function's derivative is equal to the function itself, then the definite integral of the function can be calculated by evaluating its antiderivative at the upper and lower limits.

  • The definite integral can be used to calculate the area between a curve and the x-axis.

  • To calculate the area between two curves, the areas under each curve are calculated separately and then subtracted from each other.

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