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Area/Reimann Sums
F.T.C. Part 1
F.T.C. Part 2
U Substitution
Miscellaneous
100
This is what a Reimann sum is, (a description and equation)
What is an approximation for the area under a curve using n rectangles of equal width. The equation for a reimann sum is: (jeopardy template will not allow)
100
What does F.T.C. part one say?
If f is continuous on [a,b] and F(x)=integral from a to x of f(t)dt, then F'(x)=f(x) on [a,b]
100
What does the F.T.C part 2 say?
If f is continuous on [a,b] and F(x) is any antiderivative of f, then the integral from a to b of f(x)dx = F(b)-F(a).
100
Use u-substitution to find the integral of the function : (teacher will give)
from teach
100
How can you find the average value of the area under a curve from 0 to b?
Divide the value of the integral from 0 to b by b. or: (1/b)*integral from 0 to b.
200
Let f be a continuous function on the closed interval [0,5]. If 2
Greatest - 6x5 = 30 Least - 2x5 = 10
200
If F(x)=the integral from 3 to x of (2tsquared +7t-sin(t)) dt, what is F'(x)?
2xsquared +7x-sin(x)
200
If F and f are continuous functions so that F'(x)=f(x) for all x, then the integral from a to b of f(x)dx is =
F(b)-F(a)
200
Use u-substitution to find the integral of the function : (teacher will give)
from teach
200
A bug is crawling up a wire. Its velocity at any time t is given by the function y=2x when 0
Changes direction at time 2, total distance traveled is 3.
300
The closed interval [a,b] is partitioned into n equal sub intervals, .......... from teach
from teach
300
If F(x) = the integral from 0 to xcubed of (e ^(2t)) dt, find F'(x)
e^(2xcubed) times 3 x squared.
300
Determine the area under the curve of 1/xsquared from 1 to 2.
1/2
300
Use u-substitution to find the integral of the function : (teacher will give)
300
If f(x)=g(x)+5 for 2
2 times integral of g(x) dx + 15
400
The graph of f will be shown by teacher. let the function g(x) = integral from a to x of g(t)dt. Determine for what value of x does g(x) have a maximum?
c- area after c is negative, so it decreases the value of g(x) which is the area under the curve from a to x.
400
If F(x) = the integral from 0 to x of the cubed root of (t squared +2) dt, find F'(5)
cubed root of 27 = 3.
400
Determine the area between the curve y=4-xsquared and the x axis between x=0 and x=3.
23/3
400
Use u-substitution to find the integral of the function : (teacher will give)
from teach
400
Determine the (positive) value of k given that the integral from 0 to k of 12k(squared)x - 20x(cubed)dx = 16
k=2
500
The table on the board shows the rate of lemonade consumption in gal/hour from a summer spree. The function for the rate of lemonade consumption is given by R(t) and is a twice differentiable and strictly increasing function. From the table on the board approximate how many gallons of lemonade were used from time 1 to 13 using a left endpoint reimann sum. Be sure to answer in the correct units. Is your approximation an under approximation an over approximation? Write an equation for the exact amount of lemonade consumed from hour 3 to hour 8 given R(t).
-402gal. -This is an under approximation because the function is strictly increasing and we used a left end point approximation. -The integral from 3 to 8 of R(t)dt
500
Explain clearly how the fundamental theorem of calculus helps to evaluate integrals and leads to the second fundamental theorem of calculus.
The first fundamental theorem of calculus says that the derivative of an integral function is the integrand function. Thus, the integral function itself is equivalent to any antiderivative of the integrand function. This leads us to the second fundamental theorem of calculus which explains how to evaluate definite integrals by finding the antiderivative, and subtracting the antiderivative evaluated from a to b - (in subtraction the constant is cancelled out.)
500
Determine the value of the definite integral from 1/sqrt(2) to sqrt(3)/2 of the function 1/sqrt(1-xsquared)dx.
pi/12
500
Use u-substitution to find the integral of the function : (teacher will give)
from teach
500
Write a function that has a value of 3 when x=2 and has a derivative of ln(sinx+5). (Hint: this does not need to be a pretty function and will contain an integral. Remember the FTC part 1!!)
integral from 2 to x of ln(sint+5)dt + 3