Anti-derivatives & FTC
Area
Riemann Sums & Definite Integrals
Numerical Integration
Substitution
100
∫(tan²x+1)dx
tanx+C
100
Evaluate Sum. ∑_(i=1)^10▒〖〖(i〗^2+i-1)〗
430
100
If: from x=2 to x=4, ∫x³dx=60, ∫xdx=6, and ∫dx=2, find ∫(x-8)dx from x=2 to x=4.
-10
100
Use Trapezoidal and Simpson's Rule. ∫_0^2▒〖x^3 dx〗 ; n=4
Trap= 4.25 Simp= 4
100
∫(x²-9)³(2x)dx
(x²-9)^4/4 +C
200
∫(x²+x+1)/(√x)dx
2/5x^(5/2)+2/3x^(3/2)+2x^(1/2)+C
200
Evaluate Sum. ∑_(i=1)^10▒〖i〖(i〗^2-1)〗
3080
200
If: from x=-1 to x=1, ∫f(x)dx=0 from x=0 to x=1, ∫f(x)dx=5, find from x=0 to x=1 of ∫f(x)dx - ∫f(x)dx from x=-1 to x=0.
10
200
Use Trapezoidal and Simpson's Rule. ∫_0^2▒〖〖√1+x〗^3 dx〗 ; n=4
Trap=3.283 Simp=3.102
200
∫(tan⁴x)(sec²x)dx
Answer: (tan⁵x)/5+C
300
∫l2x-3ldx from 0 to 3
9/2
300
Find area between the graph f(x)=x^2 and the x axis from x=0 to x=2, using rectangles of width 1/2. Use lower and upper sums as estimates.
Lower= 1.75 Upper= 3.75
300
If from x=0 to x=3, ∫f(x)dx=4 and from x=6 to x=3, ∫f(x)dx=, find ∫f(x)dx from x=0 to x=6
3
300
Use Trapezoidal and Simpson's Rule. ∫_0^8▒〖〖∛x〗^ dx〗 ; n=4
Trap=11.329 Simp=11.665
300
∫{(x)/√(1-x²)}dx
-√(1-x²) +C
400
An evergreen nursery sells a certain shrub after 6 years of growing. The growth rate during the 6 years is dh/dt=1.5t+5; where t=time in years and h=height in cm. The seedlings are 12 cm. tall when planted (t=0). Find the height after t years.
h=.75t² +5t+12
400
Use upper and lower sums to approximate area.
Lower= 12.5 Upper=16.5
400
True or False: The value of ∫f(x)dx from x=a to x=b must be positive
Answer: False ∫-dx from x=0 to x=1 is equal to -1
400
Estimate error in approx. the integral, with n=4 ∫_0^1▒〖1/(x+1) dx〗
Trap: 2 ; Error ≤ .010 Simp: 24 ; Error ≤ .001
400
You deposit $1,000 in the bank (t=0). The interest rate is dr/dt=√(2t+5). Find how much money you have after 10 years.
$1,041.67
500
A cylinder is filled with water at the rate of dV/dt=6t+.75, where t=time in minutes and V=volume in cm³. At t=0, the volume is 10 cm³. Find volume after t minutes.
V=3t²+.75t+10
500
y=-2x+3 [0,1]
-(n+1)/n +3
500
The surface area of a puddle of water spilled on the ground increases at a rate of dA/dt=t²-2, where A=surface area and t=time. Find the surface area from 0 to 10 minutes.
313 1/3 units²
500
Estimate error in approx. the integral, with n=4 ∫_0^2▒〖x^3 dx〗
Trap: 0 ; Error ≤ 0
500
The rate of depreciation dV/dt of a machine is inversely proportional to the square of t+1, where V is the value of the machine t years after it was purchased. If the initial value of the machine was $500,000, and its value decreased $100,000 in the first year, what is its value after 4 years?
$340,000
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