Since ( x^2 \ge 0 ) on ([0,2]): [ \textArea = \int_0^2 x^2 dx = \left[ \fracx^33 \right]_0^2 = \frac83 - 0 = \frac83 \ \textunits^2 ]
Whether you are trying to calculate the area under a parabola, the volume of a wine glass (a classic Zambak problem), or the work done by a variable force, the Zambak series offers a reliable, clear, and rigorous guide. For the student who feels lost in the dense forest of calculus, is the compass.
∫abf(x)dx=F(b)−F(a)integral from a to b of f of x space d x equals cap F open paren b close paren minus cap F open paren a close paren Key Integration Methods Integrals -Zambak-
The integral of ( \frac1x ) is ( \ln |x| + C ) (absolute value is necessary for negative ( x )).
To appreciate the style, consider how the book handles ( \int 2x e^x^2 dx ). Since ( x^2 \ge 0 ) on ([0,2]):
( \fracddx \left[ \int_a^x f(t) dt \right] = f(x) ).
If ( F'(x) = f(x) ), then ( F(x) ) is an of ( f(x) ). To appreciate the style, consider how the book
| ( f(x) ) | ( \int f(x) , dx ) | |---|---| | ( x^n ) (( n \neq -1 )) | ( \fracx^n+1n+1 + C ) | | ( \frac1x ) | ( \ln|x| + C ) | | ( e^x ) | ( e^x + C ) | | ( a^x ) | ( \fraca^x\ln a + C ) | | ( \sin x ) | ( -\cos x + C ) | | ( \cos x ) | ( \sin x + C ) | | ( \sec^2 x ) | ( \tan x + C ) | | ( \frac1\sqrt1-x^2 ) | ( \arcsin x + C ) | | ( \frac11+x^2 ) | ( \arctan x + C ) |
[ \int u , dv = uv - \int v , du ] Used for products of algebraic and transcendental functions.