Ch 5: The Definite Integral

Question 1

In a velocity curve, what corresponds to distance traveled?

Answer 1

You know this

Question 2

In a velocity curve, what corresponds to total displacement?

Answer 2

Area above curve - area below curve

Question 3

What is the left sum?

Answer 3

Approximates the area between f(t) and the t-axis by summing n rectangles of width

delta t = (b-a)/n

and height equal to function value at the left corner.

Question 4

What are the heights of each rectangle in a left sum?

Answer 4

f(t0), f(t1), …, f(tn)

Question 5

What is the right sum?

Answer 5

Approximates the area between f(t) and the t-axis by summing n rectangles of width

delta t = (b-a)/n

and height equal to function value at the right corner.

Question 6

What are the heights of each rectangle in a right sum?

Answer 6

f(t1), f(t2), …, f(tn)

Question 7

Under what conditions do we know that the left sum is an underestimate and right sum is an overestimate?

Answer 7

If the function is strictly increasing.

Question 8

Under what conditions do we know that the right sum is an underestimate and left sum is an overestimate?

Answer 8

If the function is strictly decreasing.

Question 9

What happens to the difference between left/right sums as you increase the number of rectangles?

Answer 9

Difference decreases

right sum - left sum | = | f(b) - f(a) | * delta t

Question 10

How do you get the absolute value of the right sum minus the left sum?

Answer 10

Subtract f(b) from f(a) and multiply the result by delta t.

Question 11

How do you solve questions about minimizing distance between left and right sums?

Answer 11

Set | R-L | to be whatever minimization distance you want in the formula for measuring distance between left and right sums.

Question 12

What is f(x) called in integral notation?

Answer 12

The integrand

Question 13

What are a and b (x values defining borders) called in integral notation?

Answer 13

“The Limits of Integration”

Question 14

Do all rieman sum subdivision need equal lengths?

Answer 14

No left and right sum are a special case

Question 15

Why is the integral from a to b = -(integral from b to a)?

Answer 15

From a to b, delta x = (a-b)/n = -(b-a)/n. Pull out the negative.

Question 16

What is the antiderivative of cos(x)?

Answer 16

sin(x)

Question 17

What is the antiderivative of -sin(x)?

Answer 17

cos(x)

Question 18

What is the antiderivative of sec^2(x)?

Answer 18

tan(x)

Question 19

What is the antiderivative of sin(x)?

Answer 19

-cos(x)

Question 20

What is the antiderivative of 1/cos^2(x)?

Answer 20

tan(x)

Question 21

What is the antiderivative of tan(x)?

Answer 21

-log(cos(x))

Question 22

What is the derivative of sin(x)?

Answer 22

cos(x)

Question 23

What is the derivative of cos(x)?

Answer 23

-sin(x)

Question 24

What is the derivative of -sin(x)?

Answer 24

-cos(x)

Question 25

What is the derivative of -cos(x)?

Answer 25

sin(x)

Question 26

What is the antiderivative of -cos(x)?

Answer 26

-sin(x)

Question 27

How do you find the average value of f(x) if f(x) is a curved line?

Answer 27

integral of f(x) from a to b / (b-a)

It’s the area under the curve divided by the change in your x value.