Estimating Distance Traveled- Integral/Antiderivative Unit

Question

An object has a velocity
v(t) = 4/t+45 ft/s. Use the interval [3, 13] with n = 5 and equal subintervals to compute the following approximations of the distance the object traveled. (Round your final answers to two decimal place.)

(a) Left-hand sum = ft

(b) Right-hand sum = ft

(c) average of the two sums = ft

Answer 1

Hi Symmeatra,

 

Using sums to estimate areas, you'll use the general formula of

∑_i=1⁵ v(t_i) * Δt

 

Start by finding Δt,

Δt = (b-a)/n

Δt = (13-3)/5 = 10/5 = 2

 

You can now find the x-values of the edges of each region you're summing by starting at the left end and repeatedly adding Δt until you reach the end of your interval.

3, 5, 7, 9, 11, 13

 

When using left endpoints, you'll use the first 5 values (and not the last) as your t_i; when using right endpoints, you'll use the last 5 (and not the first).

 

a) Left endpoints:

∑_i=1⁵ v(t_i) * Δt

= v(3) * Δt + v(5) * Δt + v(7) * Δt + v(9) * Δt + v(11) * Δt

 

b) Right endpoints:

= v(5) * Δt + v(7) * Δt + v(9) * Δt + v(11) * Δt + v(13) * Δt

 

You can find each value of v(t) by plugging the various t_i into your given equation, v(t) = 4/t + 45

 

c) Add your two estimates from a) and b) and divide by 2.

 

I hope this helps. If you still have questions, please comment and ask!