AP Calc

Question 1

Integrals and anti-derivatives topics

Answer 1

1. Riemann Sums
2. Definite integrals
3. Fundamental Theorem of Calculus

Question 2

A Riemann Sum equivalent to the definite integral from a to b of f(x)dx

Answer 2

limit as delta x -> 0 of the sum of f(xi)deltax for a

a Riemann sum is an approximation:

Question 3

Given the anti derivative of any given function, what will you almost always need to add at the end?

Answer 3

C, the constant that will go to 0 once differentiated

Question 4

integral from a to b f(x)dx

Answer 4

f(b) - f(a) where F’ = f

Question 5

Derivative with respect to x of an integral from a to x? ( f(t) and dt )

Answer 5

f(x)

Question 6

d/dx of a to g(x) f(t)dt?

Answer 6

f(g(x))*g’(x)

Question 7

Given the fact that an antiderivative is the opposite direction of the norm, what must we consider?

Answer 7

We do not have the same level of certainty: the constant, C, accounts for this.

Question 8

SS signifys antiderivative: SS sec u * tan u du

Answer 8

sec u + C

Question 9

SS csc u * cot u du

Answer 9

-csc u + C

Question 10

SS tan u du

Answer 10

ln |sec u| +C

Question 11

What is the antiderivative of cot(x)?

Answer 11

1. cot(x)=cos(x)/sin(x)
2. Let t=sin(x);

=> dt=cos(x)dx.

3. Integral 1/t == ln |t|

=> Integral cot(x) = ln | sin(x) | + C

ln(|x|)+C

Question 12

Integral(square(csc(x)))

Answer 12

-cot(x) + C

Question 13

SS secu du

Answer 13

ln| secu + tanu | + C

Question 14

SS cscu du

Answer 14

-ln| csc u + cot u | +c

Question 15

SS du(sqrt (1-u²⁾ )

Answer 15

arcsin u + C

Question 16

SS du/(1+u²)

Answer 16

arctanu + C

Question 17

SS du/(u*sqrt(u² -1))

Answer 17

arcsec u u du

Question 18

SS e^u du

Answer 18

e^u + C

Question 19

SS a^u du

Answer 19

a^u /ln a

Question 20

SS du/u

Answer 20

ln |u| + C

Question 21

Displacement

Answer 21

Integral of velocity with respect to time

Question 22

Distance

Answer 22

Absolute value of the integral of velocity with respect to time:

distance is different from displacement. Cannot have negative distance.

Question 23

Area between f and g, where f(x) >= g(x) from a to b

Answer 23

SS from a to b of ( f(x) - g(x) ) dx

Question 24

For any problem dealing with change:

If R(t) represents a rate of change, then the SS fomr a to b R(t)dt represents…

Answer 24

the accumulated amount of change during the timer interval from a to b.

MPH ==> total miles driven (minus negative displacement)

10 gals/sec ==> how many gallons are in the pool

etc etc

Question 25

Volume by disks

Answer 25

1. Rotate function about x-axis- dx rectangles become dx-height cylinders e.g "disks" 2. Same approximating model- limit as dx goes to 0 of sum of disks 3. Volume of each disk ==\> pi\*r²\*h = pi \* ( f(x) )² \* dx 4. SS pi\*f(x)² dx

Question 26

Volume by washers

Answer 26

dV = pi(R² - r²) dx

Question 27

How do you compute the Left Riemann Sum of a function?

Answer 27

For the left Riemann sum, approximating the function by its value at the left-end point gives multiple rectangles with base Δx and height f(a + iΔx). Doing this for i = 0, 1, ..., n − 1, and adding up the resulting areas gives: ![]()

Question 28

The left Riemann sum amounts to an overestimation if f is ?

Answer 28

The left Riemann sum amounts to an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing. ![]()

Question 29

What is the Left Riemann Sum of a function?

Answer 29

* Let ![]() * If ![]() for all *i*, then *S* is called a **left Riemann sum**.

Question 30

Draw a diagram of the Left Riemann sum of *x*³over [0,2] using 4 subdivisions

Answer 30

Left Riemann sum of *x*³over [0,2] using 4 subdivisions [![]()](https://en.wikipedia.org/wiki/File:LeftRiemann2.svg) [![]()](https://en.wikipedia.org/wiki/File:LeftRiemann2.svg)

Question 31

Draw a diagram of the right Riemann sum methods of *x*³over [0,2] using 4 subdivisions

Answer 31

Riemann sum methods of *x*³over [0,2] using 4 subdivisions [![]()](https://en.wikipedia.org/wiki/File:RightRiemann2.svg) Right

Question 32

Riemann sum methods of *x*³over [0,2] using middle subdivisions

Answer 32

USE MRAM

Question 33

Draw a diagram of Riemann sum methods of x³over [0,2] using trapezoidal subdivisions.

Answer 33

Riemann sum methods of x3over [0,2] using trapezoidal subdivisions ![]()

Question 34

Compute the area under the curve of *y* = *x*² between 0 and 2 using Riemann's method.

Answer 34

1. The interval [0, 2] is firstly divided into *n* subintervals, each of which is given a width of ![](); these are the widths of the Riemann rectangles (hereafter "boxes"). 2. Because the right Riemann sum is to be used, the sequence of *x* coordinates for the boxes will be ![](). 3. Therefore, the sequence of the heights of the boxes will be ![](). 4. It is an important fact that ![](), and ![](). 5. The area of each box will be ![]() and therefore the *n*th right Riemann sum will be: * ![]()

1. Take the limit as *n* → ∞ 2. ![]()

This method agrees with the definite integral as calculated in more mechanical ways: * ![]()

Question 35

Draw the diagrams for the 4 Riemann Sum methods for an arbitrary function f.

Answer 35

[![]()](https://en.wikipedia.org/wiki/File:Riemann_sum_convergence.png) Four of the Riemann summation [methods](https://en.wikipedia.org/wiki/Riemann_sum#Methods) for approximating the area under curves. 1. **Right** and **left**methods make the approximation using the right and left endpoints of each subinterval, respectively. 2. **Maximum** and **minimum** methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. 3. The values of the sums converge as the subintervals halve from top-left to bottom-right.