Chapter 4

Question 1

Reimman theorem

Answer 1

RAM

Question 2

if g(x) = (integral with bottom bound at pi, upper at pi*x) cos(t^2)dt, then g’(x) =

Answer 2

pi cos (pi^2 x^2)

chain rule the pi in the upper bound

Question 3

define the reverse integral property….

Answer 3

if you switch the bounds, the integral is multiplied by -1

Question 4

say you have a function f’(x). what function would represent the integral? explain.

Answer 4

f(x), as the integral is a function of what exists under the curve, or the product of the axes. thought of in terms of physics, the velocity function, being the derivative of distance, can be changed back to distance if multiplied by time. similarly, acceleration plotted against time can be multiplied to cancel out an s in the m/s^2, creating velocity.

Question 5

region A (under f(x) has an area of 1.5, from 0 to 6 the integral of f(x) = 3.5. region a is under the curve, and the only other region, region b, is above the curve. find:

1. integral from 2 to 6 of f(x)
2. integral from 0 to 6 of absolute value of f(x)
3. integral from 2 to 6 of (2 + f(x))

Answer 5

1. 5, as total area = region a + region b, thus 3.5 = -1.5 + region b
2. same as above but region a is positive (abs value)
3. split the integral into integral of 2 + integral of f(x), and solve from there (straight line at 2 means just multiply 2 by delta x

Question 6

given integral from 1 to 4 of f ‘ (x) dx = 6.2, and f(1) = 3, find f(4)

Answer 6

the integral means essentially f(x) (antiderivative cancels out the prime) from 1 to 4 = 6.2. thus, 6.2 = f(4) - f(1), and thus 6.2 = f(4) - 3.

Question 7

suppose R(t) = 3t - 18 is the rate, measured in gallons per minute, at which water is leaking out of a container.

a. calculate integral from 0 to 6 R(t) dt

b. what are the units of the above?

c. what is the meaning of the integral in part a in the context of the problem?

d. suppose there were 100 gallons of water in the tank initially. how many gallons are there in the tank after 6 minutes?

Answer 7

a. take antiderivative of 3t - 18, which = (3/2)*t^2 - 18t, then plug in 6 and 0 such that the value at 0 is subtracted from the value at 6 (upper - lower bounds). the answer is -54 gallons

b. gallons

c. that there are 54 gallons lost from the container between minutes 0 and 6

d. 46 gallons (100-54)

Question 8

what is the definite integral of a constant?

Answer 8

the constant times the difference between bounds

Question 9

when checking for absolute maxes/mins, what is the most important step that ppl forget? (final step)

Answer 9

creating a table of values with critical numbers AND endpoints and testing all

Question 10

what axis should your bounds be when going for an area between a top and bottom curve? when right to left?

Answer 10

top-bottom: x-axis

right-left: y axis

Question 11

AP CLASSROOM QUESTION 9 BY HAND?????

Answer 11

HOW

Question 12

ap classroom question 16 and 20

Answer 12

bro plzzz rmr that you can split up integrals if added/subtracted into constants multiplied by change in bound and the variable stuff on the other side

Question 13

from bounds 0 to x, what is the integral of sinx?

Answer 13

1-cosx