Calculus Problem

Question

A yeast culture is growing at a rate of  grams per hour where t is time in hours since the culture was initiated. Use Riemann Sums with 50 left-point rectangles to compute and then interpret the area under Y'(t) on the interval from t=0 to t=10.

Answer 1

The left Reimann Sum (also known as the left endpoint approximation) uses the left endpoints of a subinterval:

∫^a_b Y'(t) dt ≈ Δt (Y'(t₀) + Y'(t₁) + Y'(t₂) + ... + Y'(t_n-1))

where Δt = (b-a)/n

We have that a=0, b=10 and n=50

Therefore,  Δt = (b-a)/n = (10-0)/50 = 1/5

Divide the interval [0,10] into n=50 subintervals of the length  with the following endpoints: a = 0, 1/5, 2/5, 3/5, ..., 48/5, 49/5, 10=b

Now just evaluate Y'(t) at the left endpoints of the subinterval

Y'(0) = 0.3e^0.1(0) =0.3

Y'(1/5) = 0.3e^0.1(1/5) ≈ 0.306060402008

...

Y'(49/5) = 0.3e^0.1(49/5) ≈ 0.799336872579

Therefore:

∫¹⁰₀ 0.3e^0.1 dt ≈ (1/5) (0.3 + 0.306060402008 +  ... + 0.799336872579) = 5.1034688575607