area and distance

Question

A ball is thrown upward at time 𝑡=7 s on planet Ubuntu, and its acceleration from time 𝑡=7 s to 𝑡=11 s is given by the function 𝑎(𝑡)=5/𝑡−6 m/s^2.

Use 4 time intervals of equal length to overestimate, and then underestimate, the relative velocity of the rising ball from time 𝑡=7 s to 𝑡=11 s.

a) overestimate: answer= what in m/s?

b) underestimate:answer=what in m/s?

Answer 1

The area under a curve is the antiderivative. Since Acceleration is the derivative of velocity, we are looking for the area under this acceleration curve. Since we are asked for an overestimate and an underestimate, we know we are looking for Left and Right Riemann sums. You don't even need to graph the curve, as long as you know what the general shape of a radical function is.

I'm assuming that you meant the acceleration equation to be a(t) = 5/(t-6) so the graph of y=5/t is shifted 6 places to the right. Your equation has the curve shifted down 6 places and would give a negative velocity.

a) so for the overestimate we will take left hand sums.

5/(7-6) + 5/(8-6) + 5/(9-6) + 5/(10-6) = 5/1 + 5/2 + 5/3 + 5/4 = 10.41 m/s

b) for the underestimate we will take the right hand sums

5/(8-6) + 5/(9-6) + 5/(10-6) + 5/(11-6) = 5/2 + 5/3 + 5/4 + 5/5 = 6.41 m/s

The actual integral is 8.04 m/s so you can see that these are indeed overestimate and underestimate.

Answer 2

If you are going to take a look at the graph of this curve from t=7 to t=11, it's going down and all the points have negative derivative. If that's the case, we can just identify the overestimate and underestimate using the Left Reimann Sum (underestimate) and Right Reimann Sum (overestimate).

(a) Left Reimann Sum formula:

∫_a^b f(t) dt = Δt (f(t₀) + f(t₁) + f(t₂) + ... + f(t_n-2) + f(t_n-1)

where Δt = (b-a)/n

f(t) = 5/t - 6, a=7, b=11, and n=4

Δt = (b-a)/n = (11/7)/4 = 1

f(t₀) = f(7) = -37/7

f(t₁) = f(8) = -43/8

f(t₂) = f(9) = -49/9

f(t₃) = f(10) = -11/2

Add them all up and multiply by Δt:

∫₇¹¹(5/t - 6)dt = 1(- 37/7- 43/8 - 49/9 -11/2) ≈ - 21.605 m/s

Negative velocity means going down.

(b) Right Reimann Sum formula:

∫_a^b f(t) dt = Δt (f(t₁) + f(t₂) + f(t₃) + ... + f(t_n-1) + f(t_n)

where Δt = (b-a)/n. (This one starts with f(t₁) and ends with f(t_n))

f(t₄) = f(11) = -61/11

∫₇¹¹(5/t - 6)dt = 1(- 43/8 - 49/9 -11/2 - 61/11) ≈ - 21.865 m/s