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RRAM
the method of approximating a definite integral over an interval using the function values at the right-hand endpoints of the subintervals determined by a partition
LRAM
the method of approximating a definite integral over an interval using the function values at the left-hand endpoints of the subintervals determined by a partition
MRAM
the method of approximating a definite integral over an interval using the function values at the midpoints of the subintervals determined by a partition
Trapezoidal Rule
use trapezoids to evaluate integrals (estimate area)
Simpson's Rule
Δx/3 (f(x₀)+4f(x₁)+2f(x₂)+4f(x₃)+...+4f(Xn-₁)+f(Xn))
Fundamental Theorem of Calculus (part 1)
Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
-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
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)?