Approximating and Computing Area and Volume
R(8) (right endpoint approximation with n=8), f(x)= 7-x, [3,5]
R(8)=5.75
Calculate the integral: ∫(0,4) x2dx
64/3
Use FTC to find the area under f(x)=x2 on [0,1]
1/3 square units
Write the integral in terms of u and du. Then evaluate: ∫(x-7)3dx
∫u3du=1/4(x-7)4+C
Evaluate the integral: ∫(6x3-9x3+4x)dx
3/2x4-3x3+2x2+C
R(6) (right endpoint approximation with n=6), f(x)= 2x2-x+2, [1,4]
R(6)=47.5
Calculate the integral of: ∫(0,3) (3t+4)dt
51/2
Find the derivative of the function g(x)=∫(0,x) √{1+t2} dt
√{1+x2}
∫ cos2 θ sin θdθ
− 1/3 cos3 θ+C
Calculate the integral: ∫(2x3-1)2dx
3/2x4-3x3+2x2+C
L(5) (left endpoint approximation with n=5), f(x)= x-1, [1,2]
L(5)≈ 0.745635
Calculate the integral of: ∫(-3,0) (2x-5)dx
-24
Evaluate using FTC: ∫(-2,0) (3x-2ex)dx
2e-2-8
Use the Change of Variables Formula to evaluate the definite integral: ∫(-1,2) √(5x+6)dx
42/5
The demand for a product, in dollars, is p=1200-0.2x-0.0001x2. Find the consumer surplus when the sales level is 500.
$33,333.33
Set up the integral (but do not compute it) to find the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by y=x4 and y = 1 about the y-axis
V = ∫(1,0) π( √ y) dy
Calculate the integral ∫(0,0.5) x cos πxdx
(π-2)/2π2
Evaluate the integral using FTC: ∫(1,4) (1/t2)dt
3/4
∫ x √ (4 − x)dx
− 8/3 (4 − x)3/2 + 2/5 (4 − x)5/2 + C
A chemical flows into a storage tank at a rate of 180+3t liters per minute, where 0≤t≤60. Find the amount of the chemical that flows into the tank during the first 20 minutes.
4,200 liters of chemical flow into the tank each minute.
M(6) (midpoint approximation), f(x)=lnx, [1,2]
M(6)≈ 0.386871
Calculate the integral ∫(1,5) ln(R)/R2 dR
4/5 − 1/5 ln 5
Calculate the integral of: ∫(-a,1) (x2+x)dx
1/3a3-1/2a2+5/6
∫ √(9-x2) /x2 dx
- √(9-x2) /x-arcsin(x/3)+C
A particle is moving along a line so that its velocity is v(t0)=t3-10t2+29t-20 ft/sec at time t. What is the displacement of the particle on the time interval 1≤t≤5?
32/3 ft