Math Flashcards
Area Between Curves Formula
How do you find the bounds if they are not given?
Set f(x) = g(x) and solve for x
Find the area of the shaded region
343/6
How do you calculate area between curves in terms of y
Integral of Right function - Left function
How do you calculate area between curves in terms of x
Integral of Top function - Bottom function
If the curves intersect at multiple points…
Split the equation into multiple integrals and add them together
Find the area of the region bounded by y = sec^2(x) and y = 8cos(x) from -pi/3 to pi/3
6*sqrt3
Find the area of the region bounded by y = x^2 and y = 4x-x^2
8/3
Set up the integral for the area of a region bounded by y = cos(pi*x) and y = 4x^2 - 1
A = 2 * integral from 0 to 1/2 of cos(pi*x)-(4x^2 - 1)dx
Find the area of the region bounded by x = y^4, y = sqrt(2-x) and y=0
22/15
Find the volume of a solid obtained by rotating the region bounded by y = x^3, y-axis, x = 0 and y = 8, about the y-axis
96pi/5
Find the volume of a region bounded by y=x and y=x^2 rotated about the x-axis
2pi/15
Find the volume of a region bounded by y=x and y=x^2 rotated around y = 2
8pi/15
Find the volume of a square pyramid with base length “l” and height “h”
1/3(l^2*h)
Volume of shell formula
What method should be used if you’re rotating around a vertical axis?
Variable x, use shell method; variable y, use disk/washer method
What method should be used if you’re rotating around a horizontal axis?
Variable x, use disk/washer method; variable y, use shell method
Find the volume of the region formed by rotating y = x and y = x^2 around the y-axis.
pi/6
Find the volume of the solid generated by rotating the region bounded by y = sqrt(x) and y = 0.
pi/2
Find the volume of the solid generated by rotating the region bounded by the x-axis and y = x - x^2 about x = 2.
pi/2
Calculus definition of work
A 200 lb. cable is hanging off the roof of a building. The cable is 100 ft. long (the building is taller). How much work is done lifting the cable to the roof.
1,000 ft. lbs. or 1355.8 joules
A force of 40 N is regained to hold a spring with a natural length of 10 cm to a length of 15 cm. How much work is needed to stretch the spring from 15 cm to 18 cm?
1.565 joules
Limit definition of average value of a function
Average value integral formula
Find the average value of f(x) = 1 + x^2 from a = -1 to b = 2
2
If f(x) is continuous on [a, b] then there exists a a<=c<=b such that f(c) = f,avg
Mean value theorem for integrals
Find the value c that satisfies the mean value theorem for f(x) = 1 + x^2 from x = -1 to x = 2
c = -1, 1
Arc length formula
Surface area formula
Function that consists of only natural numbers
Sequence
Sequence formula
sn = s1 + d(n - 1)
n = next term
d = coefficient applied to each term
Sum of the elements of a sequence
Series
A series is ________ if the sequence of partial sums is a _________ sequence. A series is _________ if it is not __________.
Convergent, convergent, divergent, convergent
If the limit of a series as n approaches infinity does not equal 0, the series ______
Diverges
Sum of a geometric series
S = a/(1 - r)
Geometric series formula
A series converges if…
The absolute value of r is less than 1
What is the term for a series that can be written as tn = an - an+1
Telescoping series
Nth partial sum formula
A series whose terms alternate between positive and negative
Alternating series
What are the 2 conditions of the alternating series test?
Alternating Series Estimation Theorem
Comparison Test Conditions
Based on the comparison test, the series converges
Root Test Conditions
NOTE: Not commonly used
Ratio test conditions
If the absolute value of a series converges, but not the original. Then it is said to be
Absolutely convergent
If the absolute value of a series diverges, but the original converges, then it is said to be…
Conditionally convergent
What test is used given these conditions?
Divergence test
What test is used given these conditions?
Geometric series test
What test is used given these conditions?
P-series test
What test is used given these conditions?
Direct comparison test
What test is used given these conditions?
Limit comparison test
What test is used given these conditions?
Alternating series test
What test is used given these conditions?
Alternating series test
What test is used given these conditions?
Root test
What test is used given these conditions?
Ratio test
What test is used given these conditions?
Telescoping series test
What test is used given these conditions?
Integral test
