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These notes are taken from my Math 141 class in Spring 1999 at Penn State - Fayette branch w/ Dr. Ivko Dimitric. The book we used was Calculus by Dennis G. Zill 3rd ed. published in 1988.
1-11-1999
Integration Techniques
´ f(x)dx - indefinite (anti-derivative) 1´² f(x)dx - definite = real #'S
Length of rectangle = ±xi; xiY = sample f(xi) x ±xi = area of rectangle I
n E f(xi) x ±xi = a´n f(x)dx i=1
y= e^ x, y'= e^ x; y= ln x, y'= 1/x
y=e^ x³ y'=3x² * e^ x³ f(x)=x³, g(x)= e^ x, e^ x³ * 3x²
chain rule: g'(f(x)) * f'(x)
y=ln v ( 1+ x² ), y'= ½ * 1/ v ( 1+ x²) * 2x = x / ( 1+ x²)
Formulas 1. loga (xy)= loga x + loga y 2. loga (x/y)= loga x - loga y 3. loga x ^ r = r * loga x
´ 1/x dx = ln x + C 1/3 ´ ( 3x² * e^ x³)dx = 1/3 e^ x³ + C
Fundamental Theorum of Calculus ´ f(x)dx = A(x), A'(x)=f(x), F'(x)=f(x), a´n f(x)dx = F(x) a¦n = F(b) - F(a)
1-13-1999
Section 5.2 Volumes By Slicing
right cylinder (not necessarily circular): solid bound by 2 congruent plane regions lying in parallel planes (bases) & lateral surface generated by moving line segment whose end pts. belong to boundaries of bases & remains perpendicular to plane of bases
V=Bh : volume of right cylinder; B: area of base
finding area of 3-D object A(x)= cross-sectional area at pt. x partition at [a,b]; n - # of subintervals
<Vk= volume cylinder=Bh <Vk= A(Xk*) * (Xk - Xk-1) <Vk= A(Xk*) * (<Xk) add all approximate values n n E <Vk= E A(Xk*) * (<Xk) approximating sum for interval k=1 k=1 n a´b f(x)dx= lim E f(Xk*) * (<Xk) approximating sum Riemann sum ÈPÈt0 k=1
ÈPÈ : norm of partition; length of longest subinterval of partition n V= lim E A(Xk*) * (<Xk)= a´b A(x)dx ÈPÈt0 k=1
ex. right circular cone w/ height h & radius R formula=1/3„r²h V= a´b A(x)dx= 0´h A(x)dx= 0´h „x²R²/h²= „R²/h² * 0´h x²dx= „R²/h² * 1/3x³ 0³h= „R²/h² * (h³/3 - 0)= „R² * 1/3h= 1/3„R²h A(x)= „R²= „(xR/h)²= „x²R²/h² r= radius partition r/R= x/h; r= xR/h
ex. base bounded by y=4-x² & x-axis V= a´b A(x)dx= -2´² A(x)dx a=[(4-x²)² * v3] /4dx
1-15-1999 class was cancelled due to weather
1-18-1999 problem cont. (a/2)² + h²=a²; a²/4 + h²=a²; 4a²/4 - a²/4=3a²/4; h=v(3a)/4; x²+y²=4²; x²+y²=16; y²=16-x²; y=v(16-x²); a=2y=2 v(16-x²); A(x)=