4.2-4.3 def integrals, area, riemann

Question 1

If f(x) goes below the x-axis, is the area and/or integral positive?

Answer 1

The area is still positive, but the integral is negative.

Question 2

When does the integral from a to a equal 0?

Answer 2

When f is defined at a

Question 3

When does the integral from b to a equal the negative of the integral from a to b?

Answer 3

When f is integrable on the closed interval from a to b

Question 4

When does the integral from c to a equal the integral from a to b plus the integral from b to c?

Answer 4

When f is integrable on the three closed intervals determined by a, b, and c, regardless of where c is at.

Question 5

When does the integral from a to b of f+g equal the integral from a to b of f plus the integral from a to b of g?

Answer 5

When f and g are integrable on the closed interval from a to b.

Question 6

When does the integral from a to b of k•f(x) equal k•the integral from a to b of f(x)?

Answer 6

When f is integrable on the closed interval from a to b and k is a constant.

Question 7

Justify when a Right sum is an over/under-approximation.

Answer 7

Since f(x) is increasing, a right sum will be an over approximation.
Since f(x) is decreasing, a right sum will be an under approximation.

Question 8

Justify when a Left sum is an over/under-approximation.

Answer 8

Since f(x) is descreasing, a left sum will be an over approximation.
Since f(x) is increasing, a left sum will be an under approximation.

Question 9

Justify when a Midpoint sum is an over/under-approximation.

Answer 9

Since f(x) is concave up, a midpoint sum will be an over approximation.
Since f(x) is concave down, a midpoint sum will be an under approximation.

Question 10

What does a velocity integral give?

Answer 10

Change in particle’s position

Question 11

How does the IVT relate upper and lower sums?

Answer 11

Lower sum < integral < Upper sum