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Level 615

Definite Integrals


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When f(x) is increasing.
When does the right Riemann sum overestimate the definite integral of f(x)?
When f(x) is decreasing.
When does the right Riemann sum underestimate the definite integral of f(x)?
When f(x) is increasing.
When does the left Riemann sum underestimate the definite integral of f(x)?
When f(x) is decreasing.
When does the right Riemann sum overestimate the definite integral of f(x)?
When f(x) is concave up.
When does the trapeziodal sum overestimate the definite integral of f(x)?
When f(x) is concave down.
When does the trapeziodal sum underestimate the definite integral of f(x)?
-b->af(x)dx
a->b f(x)dx
0
a->a f(x)dx
b->a cdx
c(b-a) where c is any constant
b->a [f(x)+g(x)]dx
b->a f(x)dx+ b->a g(x)dx
b->a cf(x)dx
c b->a f(x)dx where c is any constant
b->a [f(x)-g(x)]dx
b->a f(x)dx-b->a g(x)dx
b->a f(x)dx
c->a f(x)dx+b->c f(x)dx
then b->a f(x)dx>_0
If f(x)>_ 0 for a<_x<_b
then b->a f(x)dx>_b->a g(x)dx
If f(x) >_ g(x) for a<_x<_b
then m(b-a)<_b->a f(x)dx<_M(b-a)
If m<_f(x)<_M for a<_x<_b
∫ c(fx)dx
c∫ f(x)dx
∫kdx
kx+C
∫(e^x)dx
e^x+C
-cos(x)+C
∫ sin(x)dx
tan(x)+C
∫ sec^2(x)dx
sec(x)+C
∫ sec(x)tan(x)dx
tan^-1x+C
∫ 1/(x^2+1)dx
ln|x|+C
∫ (1/x)dx
∫cos(x) dx
sin(x)+C
-cot(x)+C
Integral (csc^2(x)dx)
-csc(x)+C
∫ csc(x)cot(x)dx
sin^-1(x)+C
∫ 1/sqrt(1-x^2)dx