Science One Integral Calculus January 018 Happy New Year!
Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)?
Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)? rate of change of f (with respect to x) at x (geometrically) slope of tangent line to graph of f at x What is the mathematical definition of f (x)?
Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)? rate of change of f (with respect to x) at x (geometrically) slope of tangent line to graph of f at x What is the mathematical definition of f (x)? a limit! lim *, f(x+h) f(x) * or equivalently -/ lim -., -.
Integral Calculus central idea: The Definite Integral What is the definite integral f x dx 3?
Integral Calculus central idea: The Definite Integral What is the definite integral f x dx? 3 (geometrically) area of region under curve above [a, b] (only if f(x) > 0 on [a,b]) (depending on context) other interpretations e.g. if f = v(t) velocity then the definitive integral v t dt 3 distance traveled in time interval Δt = b a is the What is the mathematical definition of f x dx 3?
Integral Calculus central idea: The Definite Integral What is the definite integral f x dx? 3 (geometrically) area of region under curve above [a, b] (only if f(x) > 0 on [a,b]) (depending on context) e.g. if f = v(t) velocity then the definitive integral v t dt 3 distance traveled in time interval Δt = b a is the What is the mathematical definition of f x dx 3? a limit!
(some of) our goals this term Formulate a precise definition of the definite integral Establish a fundamental connection with the derivative (Fundamental Theorem of Calculus) Master integration techniques to compute complicated antiderivatives Apply concept of integration to a variety of science contexts Today s goal: Formulate a precise definition of the definite integral.
The area problem: Find the area of the region S that lies under the curve y = f(x) from a to b. What is area? Easy for regions with straight sides. Not so easy for regions with curved sides. need a precise definition of area.
Example: Find the area under f(x)=x on [0,1]. [worksheet] We found that the sum S n of areas of n rectangles converges as n à We define area S as a limit: S = lim D E S D
The definite integral Consider the region under the curve y = f(x) above [a, b]. Take n vertical strips of equal width Δx = (b a)/n Consider n intervals: [x 0,x 1 ],[x 1, x ],[x, x 3 ], [x MNO, x i ], [x DNO, x n ].
The definite integral Consider the region under the curve y = f(x) above [a, b]. Take n vertical strips of equal width Δx = (b a)/n Consider n intervals: [x 0,x 1 ],[x 1, x ],[x, x 3 ], [x MNO, x M ], [x DNO, x n ]. Sum areas of all rectangles D MSO S n = Δx f(x 1 *) + Δx f(x *) +. + Δx f(x i *) + + Δx f(x n *) = f(x M ) where sample point x i * is any number in the interval x MNO, x M. Δx
The definite integral Consider the region under the curve y = f(x) above [a, b]. Take n vertical strips of equal width Δx = (b a)/n Consider n intervals: [x 0,x 1 ],[x 1, x ],[x, x 3 ], [x MNO, x M ], [x DNO, x n ]. Sum areas of all rectangles S n = Δx f(x 1 *) + Δx f(x *) +. + Δx f(x i *) + + Δx f(x n *) = D MSO f(x M ) Δx where sample point x i * is any number in the interval x MNO, x M. Defn: The area S of the region under the graph of f above [a, b] is D S = lim S n = lim f(x D E D E MSO M ) Δx = 3 f x dx
The definite integral Consider the region under the curve y = f(x) above [a, b]. Take n vertical strips of equal width Δx = (b a)/n Consider n intervals: [x 0,x 1 ],[x 1, x ],[x, x 3 ], [x MNO, x M ], [x DNO, x n ]. Sum areas of all rectangles S n = Δx f(x 1 *) + Δx f(x *) +. + Δx f(x i *) + + Δx f(x n *) = D MSO f(x M ) Δx where sample point x i * is any number in the interval x MNO, x M. Defn: The area S of the region under the graph of f above [a, b] is D S = lim S n = lim f(x D E D E MSO M ) Δx = 3 f x dx Definite Integral
The definite integral Consider the region under the curve y = f(x) above [a, b]. Take n vertical strips of equal width Δx = (b a)/n Consider n intervals: [x 0,x 1 ],[x 1, x ],[x, x 3 ], [x MNO, x M ], [x DNO, x n ]. Sum areas of all rectangles S n = Δx f(x 1 *) + Δx f(x *) +. + Δx f(x i *) + + Δx f(x n *) = D MSO f(x M ) Δx where sample point x i * is any number in the interval x MNO, x M. Defn: The area S of the region under the graph of f above [a, b] is D S = lim S n = lim f(x D E D E MSO M ) Δx = 3 f x dx Riemann Sum