advertisement

5.4 THE FUNDAMENTAL THEOREM OF CALCULUS 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 b f ( x)dx F (b) F (a) a **F(b) – F(a) is often denoted as F ( x) b a This part of the FTC is significant because it allows us to evaluate definite integrals. FTC…THE PART THAT CONNECTS IT ALL… x If f is continuous on [a, b] and F ( x) f (t )dt a then F’(x) = f(x) at every point x in [a, b]. You may also see this as x dF d d f (t )dt f ( x) dx dx a dx WHAT’S THE SIGNIFICANCE? Every continuous f is thex derivative of some other function, namely f (t )dt a Every continuous function has an antiderivative. The processes of integration and differentiation are inverses of each other! EXAMPLE If x F ( x) (t 2 4)dt , integrate to find F(x). Then, 1 differentiate to find F’(x). x 1 3 F ( x) t 4t 3 1 1 1 x3 4 x 4 3 3 1 3 11 x 4x 3 3 F ' ( x) x 4 2 EXAMPLE Using FTC for x F ( x) (t 2 4)dt , 1 F ' ( x) x 4 2 find F’(x). EXAMPLE If x h( x) t 1dt , find h’(8). 3 h' ( x ) x 1 h' (8) 8 1 3 EXAMPLE 1 1 dy Find if y 3 dt t dx 2x 2x 1 y 3 dt t 1 1 2 3 (2 x) 1 3 4x (change bounds) EXAMPLE Construct a function in the form of x y f (t )dt C a that has tan x as the derivative and satisfies f(3) = 5. x dy tan tdt tan x and dx 3 x x tan tdt 0 3 y tan tdt 5 3 when x = 3, so TOTAL AREA VS. NET AREA 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.” FINDING TOTAL AREA ANALYTICALLY To find the area between the graph of y = f(x) and the x-axis over the interval [a, b] analytically, 1. Break [a, b] apart using the zeros of f 2. Integrate f over each subinterval 3. Add the absolute values of the integrals EXAMPLE Find the area of the region between the curve y = 4 – x2 and the x-axis over [0, 3]. 1. Break the curve into two subintervals since part of it is above the x-axis and the other is below. 2. Break the integral into [0, 2] and [2, 3] 2 3 2 2 ( 4 x ) dx ( 4 x )dx 0 (subtract to make the second area positive) 2 2 3 x x 4 x 4 x 3 0 3 2 3 3 16 7 23 3 3 3 EXAMPLE Find the net area of the region between the curve y = 4 – x2 and the x-axis over [0, 3]. 3 2 ( 4 x )dx 0 3 x 4 x 12 9 3 3 0 3 TOTAL AREA USING FINT To find the area between the graph of y = f(x) and the x-axis over the interval [a, b] numerically, evaluate b a f ( x) dx