Antiderivatives

Antiderivatives

1. Concepts

   • Substitution Method
     • ginagamit para maging mas 'madali' yung pag-iintegrate, basically, ginagawa mong similar sa mga usual antiderivatives (see list of antiderivatives) yung itsura ng problem mo.
     • (1) let u= some function such that yung du mo is makukuha mo sa natitirang function sa loob ng integral. (2) find du. (3) write your integral in terms of u and du na lang. (4) solve the 'easier' integral.
     • may be used more than once sa solution; pwedeng gamitin hanggang maging simple na lang yung integral
     • galing sa chain rule sa derivatives.
   • Fundamental Theorem of Calculus
     • FFTC
       • If F(x) = int_ c^ x f(t) dt, then F'(x)=f(x)
       • If F(x) = int_ c^(g(x)) f(t) dt, then F'(x)=f(g(x)) g'(x)
     • SFTC
       • If F(x)= int f(t) dt, then int_a^b f(t) dt = F(b) - F(a)
   • Definite Integrals
     • area under the curve
     • utilizes SFTC
     • may limits of integration, of the form int_a ^b f(t) dt
   • General vs Particular Antiderivatives
     • General antiderivatives - may + C, walang condition
     • Particular antiderivatives - kapag may initial or boundary condition. Ex. find the antiderivative F of f, given that F(0)=c.
       • (1) Hanapin si General Antiderivative, (2) plug in si initial or boundary condition as sagot (3) get the value of C (constant of integration)
       • initial condition : given ang value ng function at t=0.
       • boundary condition, given ang value ng function sa endpoints ng some interval. ex. f(a) at f(b)
   • integrability
     • if differentiable, then continuous, which implies integrable
     • continuity implies integrability

2. Application

   • Rectilinear Motion
     • application ng particular antiderivatives
     • if given a(t) : (1) find integral v = int a(t) dt (2) use condition to get v(t)
     • if given v(t): (1) find integral s(t) =int v(t) dt (2) use condition to get s(t)
     • object starts from rest: v(0) = 0
     • object at ground: s(t)=0
     • max height attained: v(t) = 0
   • Area
     • using horizontal rectangles
       • all functions should be in terms of y
       • int_c ^d (right - left) dy
     • using vertical rectangles
       • all functions should be in terms of x
       • int_a ^b (upper - lower) dx
     • (1) diagram + find intersection points (2) choose if horizontal or vertical rectangles (3) find upper (right) and lower (left) functions and respective intervals and divide into subregions if necessary (4) make sure functions are in correct variable (5) plug in to the correct formula, with correct limits of integration
     • when rewriting functions in terms of the other variable, for example, from y= f(x) to x=g(y), and the function involves y^2, make sure to identify if the graph uses the upper half or lower half of the graph to determine the sign of the sqrt.
       • Ex. x^2+y^2=1, tapos lower half lang ng function and in terms of y yung kailangan, then use x = -sqrt(1-y^2)
   • Arc Length
     • for finding perimeter of a region, or just a part of the boundary of the region
     • f is smooth on [a,b] if f' is continuous on [a,b]
     • if looking for perimeter of the region, find the arc length of each boundary.
     • (1) identify the interval (2) find dy/dx and make sure that it is smooth in the interval. If not, find dx/dy. (3) plug into the formula.
     • int_a ^b sqrt(1+ (dy/dx)^2) dx or int_a ^b sqrt(1+ (dx/dy)^2) dy
       • dy/dx is derivative in terms of x, so dapat walang y
       • dx/dy is derivative in terms of y, so dapat walang x
   • Volume
     • solid of revolution
       • finding the volume (1): method of disk / washer
         • disk kapag walang butas yung solid
           • disk: int_a ^b pi (r^2) dw
         • washer pag may butas sa gitna yung solid
           • disk: int_a ^b pi (r_2^2- r_1^2) dw
         • disks and washers must be perpendicular to the axis of revolution.
           • about x = c , dy
           • about y = c, dx
         • (1) diagram (find plane region, points of intersection, revolve about the axis) (2) identify if dy or dx and the interval, you may cut the solid if necessary (3) get a sample disk/washer. identify the equation of the line that touches the boundary of your disk or washer (4) rewrite function in terms of the variable (dependent on dy or dx) (5) find r or r_1 and r_2 (6) plug into formula
         • r= eqtn of boundary ng circle - eqtn of axis of rev
       • finding the volume (2): method of cylindrical shells
       • axis of revolution = kung saan iikot yung region
       • formed by revolving a plane region about an axis of revolution
     • volume by slicing
       • (1) diagram (identify the solid, cross sections) (2) identify if dx or dy, dependent on the cross section and interval (3) find a formula for the area of the cross section (4) get a sample cross section. Identify equation to be used in the area of the cross section in terms of x or y. (5) plug into formula
       • int_a^b (area of cross section) dw
       • 'cross section'
       • sliced vertically - use dx
       • sliced horizontally - use dy

3. Review list of antiderivatives

   • polynomial x^n, n is not -1
   • trigonometric (10 items)
   • exponential a^x, a is constant
   • yielding logarithmic fcn (form ay 1/x or x^-1)
   • yielding inv circular fcn (3 items)
     • usually uses completing the square para maging of the form a^2-u^2, a^2+u^2 or u^2-a^2
   • hyperbolic (10 items)
     • may mga kamukha ng trig fcns, iba lang ang signs