Left endpoint approximation

Question

An object's velocity in meters per second after 𝑡 seconds is given by 𝑣(𝑡)= -4𝑥^2+96𝑥.

Estimate the distance the object travels from 𝑡= 0 to 𝑡= 18 using left endpoint approximation with 6 rectangles.

Answer 1

Perhaps a sketch might help:

Each rectangle is 3 units wide. The height of each rectangle is f(left point) so the height of the 1st rectangle is f(0), the height of the 2nd is f(3), the height of the 3rd is f(6), the height of the 4th is f(9) the height of the 5th is f(12) and the height of the 6th is f(15).

The area of each rectangle is its width times its height.

Since f(0) = 0, the area of the 1st rectangle is (3)•(0) = 0

Since f(3) = 252, the area of the 2nd rectangle is (3)(252) = 756

and so forth.

Add the areas of the 6 rectangles up to get the total.

Answer 2

Letf endpoint & 6 rectangles

n=6

a=0, b=18 => delta x = (18-0) / 6 = 3.

x1 = a = 0 (since we are using left enpoint)

x2 = a+ delta x = 0+3 = 3

x3 = 3+3 = 6

x4 = 6+3 = 9

x5 = 9+3 = 12

x6 = 15

f(x1) = f(0) = 0

f(x2) = f(3) =-4 * 3^2 +96*3 = 252

f(6) = 432

f(9) = 540

f(12) = 576

f(15) = 540

Est Distance = 3*f(x1) + . . . +3*f(x6)

= 0*3 + 252*3+432*3+ ... + 540*3 = 7020