Unit Four: Integrals

Question 1

What is the indefinite integration of {0dx?

Answer 1

C

Question 2

What is the indefinite integration of {kdx?

Answer 2

Kx + c

Question 3

What is the indefinite integration of {k f(x) dx?

Answer 3

K {f(x) dx

Question 4

What is the indefinite integration of {f(x)+/-g(x)dx?

Answer 4

{f(x)dx +/- {g(x)dx

Question 5

What is the indefinite integration of {x^ndx?

Answer 5

1/(n+1) x^(n+1) + c

Question 6

What is the summation of c if if i starts at 1 and goes to n?

Answer 6

c * n

Question 7

What is the summation of i if if i starts at 1 and goes to n?

Answer 7

n*(n+1) divided by 2

Question 8

What is the summation of i^2 if if i starts at 1 and goes to n?

Answer 8

n(n+1)(2n+1) divided by 6

Question 9

What is the summation of i^3 if if i starts at 1 and goes to n?

Answer 9

n^2*(n+1)^2 divided by 4

Question 10

How do you find the area under a curve (sums)?

Answer 10

Find the infinite sum of rectangles infinitely small in width

Question 11

How do you change a sum of infinite rectangles into an integration?

Answer 11

Integration from first x to second x of the function

Question 12

If f is defined at a, what is the integration from a to a of a function?

Answer 12

0, there is no width

Question 13

If f is defined on [a,b] then what is the relationship between the integration from b to a of a function and the integration from a to b of the same function?

Answer 13

The second is negative of the first

Question 14

How do you integrate an absolute value?

Answer 14

Break it apart into two integrations with two different functions

Question 15

What is the fundamental theorem of calculus? MEMORIZE

Answer 15

If a function f is continuous on [a,b] and F is an antiderivative of f on [a,b], then the integration of f(x) from a to b is equal to F(b)-F(a)

Question 16

What is the mean value theorem for integrals? MEMORIZE

Answer 16

If f is continuous on [a,b], then there exists a number c in [a,b] such that the the integration of f(x) from a to b is equal to f(c)(b-a)

Question 17

What is the average value of a function?

Answer 17

If f is integratabtle on [a,b], then the average value of f on the interval is 1/(b-a) times the integration of f(x) from a to b

Question 18

What is the second fundamental theorem of calculus? MEMORIZE

Answer 18

If f is continuous on an open interval I containing a, then for every x in I the derivative of the integration of f(t) from a to x is f(x)

Question 19

What is the integration of f(g(x))g’(x)dx?

*It is equal to F(g(x))+C

Answer 19

If u is equal to g(x), then du is equal to g’(x)dx then the integration of f(u)du is F(u) + C

Question 20

What is the integration of f(g(x))g’(x)dx from a to b?

Answer 20

If u is equal to g(x), then du is equal to g’(x)dx then it is equal to the integration of f(u)du from g(a) to g(b)

Question 21

What is the formula for trapezoidal approximations of the area under a curve?

Answer 21

(b-a)/n times 1/2 times (f(x0) + 2f(x1) +2f(x2) + … + 2f(Xn-1) + f(xn))

Question 22

What is the formula for Simpson’s rule and what is the requirement?

Answer 22

(b-a)/n times 1/3 times (f(x0) + 4f(x1) +2f(x2) + 4f(x3) + 2f(x4) + … + 4f(Xn-1) + f(xn)), n is even

Question 23

Derivative of sin

Answer 23

Cos

Question 24

Derivative of cos

Answer 24

-sin

Question 25

Derivative of tan

Answer 25

Sec^2

Question 26

Derivative of csc

Answer 26

-csc*cot

Question 27

Derivative of sec

Answer 27

Sec*Tan

Question 28

Derivative of cot

Answer 28

-csc^2