Areas and Distances
Definite Integrals
Indefinite Integrals
Fund. Thm.
of Calculus
of Calculus
Substitutions
100
The estimated area using four approximating rectangles and right endpoints under the graph
f(x)=cos(x)
from
x=0 \text{ to } x=\pi/2
What is 0.790766260123 or
pi/8(cos(pi/4)+cos([3pi]/8)+sqrt2 /2)
100
The answer to \int_0^5f(x)dx when
f(x)={(3,ifx<3),(x, if x>=3 ):}
what is 9?
100
The general antiderivative of \intx^n\ dx with
n\ne -1.
What is
1/(n+1) x^(n+1) +C ?
100
What is the derivative of the function:
int_1^x (2+t^4)^5 dt
What is
(2+x^4)^5 ?
100
The u substitution and integral set up to the problem
int_0^4 x+1(x^2+2x+3) dx ?
What is
u=x^2+2x+3 and 1/2int_3^27u\ du ?
200
The estimation of the area (use a trapezoidal Riemann sum) under the graph f(x)=1+x^2, -1 \leq x \leq 1
with n=4
What is 2.75?
200
The answer the the integral
int_-3^3 (x+2) dx ?
What is 12?
200
The general antiderivative of
\int sec(x) tan(x) dx
What is
sec(x) +C ?
200
The derivative of the function
int_0^(x^2) sqrt(1+r^3) dr
What is
2x sqrt(1+x^6) ?
200
The solution to the definite integral
int_0^1 sin(3pit)dt
What is
2/(3pi) ?
300
Find an Expression for the area under the graph f as a limit
f(x)=(2x)/(x^2 +1), 1\leq x \leq 3
What is
lim_(n\rightarrow \infty) \sum_{i=1}^n (2/n)((2x_i)/(x_i^2+1))
300
The answer to the integral:
int_9^9 (tan^2(x) *sin(x)) \ dx .
What is 0?
300
The general antiderivative of \int sin(x) dx
What is
-cos(x)+C ?
300
The interval in which the following curve is concave downward y=int_0^x f(t) dt with
f(t)= 1/(t^2 +3t+2) .
What is the interval (-infty, -3/2) ?
300
The answer to the integral of the following problem : int usqrt{1-u^2} du
What is
-1/3 (1-u^2)^(3/2) + C ?
400
The limit of a Riemann sum is the version to this integral
int_0^(2\pi) x^2 sin(x) dx
What is lim_(nrightarrow infty) sum_(i=0)^n (2pi)/(n)(x_ i)^2sin(x_i) ?
400
The answer to the integral 
int_0^7 g(x) \ dx.
What is
4 1/2 -2pi?
400
The general antiderivative of
\int(x^2+x^-2)dx
What is
1/3 x^3 -1/x +C ?
400
The interpretation of int_0^2 w'(t)\ dt where w'(t) is the weight of a baby's growth in pounds per week.
What is the increase of the baby's weight from zero to the second week?
400
Evaluate the integral: int \ sec^2 (1/x) /x^2 dx, u=1/x
What is
-tan(1/x)+C ?
500
AP Classroom Question Processes Riemann Sums #7
C
500
We can say this because of the following statement:
1) f(x) >= g(x) for a<=x<=b
What is the area under f is greater or equal to the area under g?
What is
int f(x)\ dx >= int g(x)\ dx?
500
The general antiderivative of
\int \ (sin2x)/sinxdx
What is
2 sin(x) +C?
500
The average value of the function f(x)=x-x^2 , [0, 2] .
what is
f_{avg}=f(c)=-5/3?
500
The solution to the integral int_0^5 (x^4 -8x+7)dx
What is 660?