AP Calculus Chapter 5

integral of secx

ln|secx+tanx|+C

integral of cotx

ln|sinx|+c or -ln|cscx|+C

a is the ______________ limit on an integral

lower

what makes an integral definite?

it has an upper and lower limit

how do you find the length of each subinterval?

(b-a)/n

for indefinite integrals, what must be added on at the end? what does it signify?

+c, for any constant

integral from b to a f(x)dx is equal to

- integral from a to b f(x)dx

integral of tanx

-ln|cosx|+C or ln|secx|+C

integral of cscx

-ln|cscx+cotx|+c

in the calculator, the percent for a left riemann sum is ________

0

integral from -a to a of an odd function

0

average value of a function

1/(b-a) (integral from a to b f(x)dx)

in the calculator, the percent for a right riemann sum is _______

100

integral from -a to a of an even function

2 integral from 0 to a of the function

in the calculator, the percent for a midpoint riemann sum is ________

50

Trapezoidal Approximation

A=(h1/2b(1)+b(2)+b(3)...+1/2b(last))

Riemann Sum

Splitting the area under a curve into rectangles in which you can find the sum of all the areas to approximate the area of the region under the curve

midpoint riemann sum

divide the interval into the required amount of subintervals, choose the center point in each subinterval to determine the height of the rectangles, find the areas and add them together

left riemann sum

divide the interval into the required amount of subintervals, choose the farthest point to the left in each subinterval to determine the height of the rectangles, find the areas and add them together

right riemann sum

divide the interval into the required amount of subintervals, choose the farthest point to the right in each subinterval to determine the height of the rectangles, find the areas and add them together

F indicates the ______________

integrated function

Substitution rule for indefinite integrals

substitute a value/equation out of an integral for the variable u and put in terms of du

When the area trapped between the curve and the x-axis is both above and below the x-axis, the area below must be ___________________ the area on top to get the total

subtracted from

how to find du

take the derivative of the value/equation you plugged u in for

how to interpret sigma notation

the lower number equals the lowest x value and the top number is the highest x value; the function after the sigma sign indicates what you plug each value into; the sigma itself indicates to plug in each x value into the function from the lowest to the highest and add them all together

what does n indicate?

the number of subintervals the interval is divided into

MVT for integrals

there exists a point c in a continuous interval at which f(c) = average value of the function

b is the ________________ limit on an integral

upper

absolute error

|c-x|, where x is the exact value and c is the approximated value

relative error

|c-x|/|x|, where x is the exact value and c is the approximated value

Fundamental Theorem of Calculus

∫ f(x) dx on interval a to b = F(b) - F(a)