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5.4 The fundamental theorem of calculus

5.4 The fundamental theorem of calculus. (Part 2 of the FTC in your book) If f is continuous on [ a, b ] and F is an antiderivative of f on [ a, b ], then. the fundamental theorem of calculus. ** F(b) – F(a) is often denoted as.

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5.4 The fundamental theorem of calculus

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  1. 5.4 The fundamental theorem of calculus

  2. (Part 2 of the FTC in your book) If f is continuous on [a, b] and F is an antiderivative of f on [a, b], then the fundamental theorem of calculus **F(b) – F(a) is often denoted as This part of the FTC is significant because it allows us to evaluate definite integrals.

  3. If f is continuous on [a, b] and then F’(x) = f(x) at every point x in [a, b]. Ftc…the part that connects it all… You may also see this as

  4. Every continuous f is the derivative of some other function, namely Every continuous function has an antiderivative. The processes of integration and differentiation are inverses of each other! What’s the significance?

  5. If , integrate to find F(x). Then, differentiate to find F’(x). example

  6. Using FTC for , find F’(x). Example

  7. If , find h’(8). Example

  8. Find if Example (change bounds)

  9. Construct a function in the form of that has tan x as the derivative and satisfies f(3) = 5. example and when x = 3, so

  10. Net area counts area below the x-axis as negative area. • Computing area on an entire interval using antiderivatives helps us find net area • Total area is the entire amount of area enclosed by a graph • Problems that say find “area” from here on out imply for you to find “total area.” Total area vs. net area

  11. To find the area between the graph of y = f(x) and the x-axis over the interval [a, b] analytically, Break [a, b] apart using the zeros of f Integrate f over each subinterval Add the absolute values of the integrals Finding total area analytically

  12. Find the area of the region between the curve y = 4 – x2 and the x-axis over [0, 3]. example Break the curve into two subintervals since part of it is above the x-axis and the other is below. Break the integral into [0, 2] and [2, 3] (subtract to make the second area positive)

  13. Find the net area of the region between the curve y = 4 – x2 and the x-axis over [0, 3]. example

  14. To find the area between the graph of y = f(x) and the x-axis over the interval [a, b] numerically, evaluate Total area using fint

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