General Antiderivatives + IVPS (I.4.10)
Find the antiderivative of f(x)=∫(x2+sinx)dx
(1/3)x3-cos(x)+C
For a function f(x) on the interval [2, 10] divided into 4 subintervals of equal width, this is the exact value of delta x
2
Compute ∫(2x+1)dx with bounds 1 and 4
18
∫(4/x)dx with bounds [1,e5]
20
What is the difference in the set up in a distance vs displacement problem?
Distance: need absolute values
Displacement: just integrate as is
Find the antiderivative of ∫[(6x2+5x+7)/x2]dx
6x+5ln|x|-(7/x)+c
If a function f(x) is strictly decreasing and concave up on a given interval, this specific type of Riemann sum approximation will always result in an underestimation of the true net area.
Right Riemann Sum
Compute ∫cos(x)+1 dx with bounds [0, pi]
pi
∫(8/(x2+1))dx with bounds (0, √3)
(8pi)/3
Suppose water is leaking out of a tank at a rate that is changing over time. Suppose that after t minutes the water is leaking out at a rate of R(t) gallons per minute, where R(t) = 3t2-4t+5. If the tank held 100 gallons after 1 minute, then how many gallons will be in the tank after 4 minutes?
52 gal
Find the formula for y(t) if: dy/dx = 6/t2 and y(1)=5
y=-6/t +11
This type of Riemann sum always includes the f(0) value in the area calculation
Left Riemann sum
What is the average value of the function f(x) = 4x + 1 over the closed interval [0, 3]
3
Find the critical points of the function F(x) = ∫(t2-7t+10)/(1+t4) dt on the interval [-1,x]
x=2, x=5
Find the displacement of a particle if the velocity is defined by ∫4t2+5t dt with the bounds of [2,4]
314/3
Find F that is the antiderivative of f(x) = 9sec(x)2+6csc(x)2; such that F(pi/4)=2
F(x)=9tan(x)-6cot(x)-1
Use a Riemann sum to estimate the area under the graph of f (x) = 2x over the interval [−1, 5] using 3 rectangles of equal width. Use the left endpoints of each subinterval to determine the heights of the rectangles.
12
Given: ∫h(x)dx [0,3] =4 and ∫h(x)dx [3,8] = 12. What is ∫h(x)dx [0,8]
16
Evaluate: d/dx[∫(7t+9e9t^2)dt] with the bounds [x,5]
-7x-9e9x^2
Find the total distance travelled by the particle if its velocity is described by the function v(t) = x2+2x-15 from t=0 to t=5
137/3
If a particle is moving with instantaneous acceleration of a(t) = 3t+5 m/s2; what is s(t) given v(0)=7m/s and s(0)=2m
s(t)=1/2 t3+ 5/2 t2 + 7t +2
Evaluate a Riemann sum to estimate the area under the graph of f(x) = 2^x with 3 rectangles of equal width and right endpoints on the interval [-2,4]
42
Evaluate the definite integral ∫|2x-12|dx with bounds [3,10]
25
Find a function g(x) and a value of a such that 2+∫(g(t)/t2)dt = 3x5 with bounds [a,x]
g(x) = 15x6; a=5√(2/3)
Suppose a small insect is walking along a length of rope. The acceleration of the insect (in m/s2) along the rope after t seconds is given by the formula a(t) = 2t + 6. If the initial velocity of the insect (when t = 0) is −7 meters per second, find the total distance traveled by the insect between t = 0 and t = 2
8m