Riemann Sums
Reverse that Power Rule
U-Sub
Division Square
MISC Integrals
100
Evaluate the Left Riemann Sum with equal subintervals over the interval to approximate
int_3^9f(x)dx
f(3)*3+f(6)*1+f(7)*2=
77
100
int-9dx
-9x+C
100
inte^(6x)dx
1/6e^(6x)+C
100
intdx/(4+(x-1)^2)
1/2arctan((x-1)/2)+C
100
int(sec^2(x)+cos(x)+1)dx
tan(x)+sin(x)+x+C
200
Evaluate the Right Riemann Sum using the table values below to approximate
3int_2^6f(x)dx
3(f(6)+f(5)+f(4)+f(3))
=45
200
int7/x^2dx
-7/x+C
200
int6x^2sqrt(3x^3-1)dx
4/9(3x^3-1)^(3/2)+C
200
int(x^3-3x+2)/(x+1)dx
1/2x^2-4x+6ln|x+1|+C
200
int(3e^x+1/x-1/x^2)dx
3e^x+ln|x|+1/x+C
300
Estimate the area underneath the curve for the following function using midpoint Riemann Sum from x=0 to x=4 with 2 equal subintervals
f(x)=x^3+4
(f(1)+f(3))*2
(5+31)*2
72
300
int(x-5)^2dx
1/3x^3-5x^2+25x+C
300
int5/(xln(x))dx
5ln|ln|x||+C
300
intdx/(x^2-2x+2)
arctan(x-1)+C
300
Given that
f(x)=int_-1^(x^2)sqrt(t^3+3)dt
find f'(1)
4
400
Approximate the area under f(x) on the interval [0,4] using Right Riemann Sum with equal subintervals given
f(x)=4x-x^2
10
400
int(x^3-4x-1)/(2x^3)
1/2x+2/x+1/(4x^2)+C
400
int3/(sqrt(9-4x^2))dx
3/2arcsin((2x)/3)+C
400
int(x^3-3x^2+5)/(x-3)dx
x^3/3-5ln|x-3|+C
400
ANIME CLUE: The signature ability of Yusuke Urameshi from the anime Yu Yu Hakusho
Spirit Gun :)
500
What is the trapezoidal sum approximation for
int_4^14f(x)dx
using three subintervals as indicated by the table
189/2 or 94.5
500
int(3x-2)^2/sqrt(x)
18/5x^(5/2)-8x^(3/2)+8x^(1/2)+C
500
intsqrt(cos(6x))sin(6x)dx
-1/9(cos(6x))^(3/2)+C
500
int1/sqrt(-x^2-x)dx
arcsin((x+1/2)/(1/2))+C
arcsin(2x+1)+C
500
Let f(x) be an odd function that is integrable on the interval [-a,a], then what is the value of the integral below. Explain your reasoning (Hint think of the cubic function)
int_-a^af(x)dx
Odd functions are symmetrical across the origin, thus creating areas of equal size but opposite sign, thus they cancel out and the net area will be 0.