How do you evaluate the definite integral by the limit definition given #int x/2dx# from [0,4]?

Question

Elijah Abram · Accepted Answer

Please see the explanation section below.

Here is a limit definition of the definite integral. (I hope it's the one you are using.) I will use what I think is somewhat standard notation in US textbooks. .#int_a^b f(x) dx = lim_(nrarroo) sum_(i=1)^n f(x_i)Deltax#. Where, for each positive integer #n#, we let #Deltax = (b-a)/n# And for #i=1,2,3, . . . ,n#, we let #x_i = a+iDeltax#. (These #x_i# are the right endpoints of the subintervals.) I prefer to do this type of problem one small step at a time. #int_0^4 x/2 dx#. Find #Delta x# For each #n#, we get #Deltax = (b-a)/n = (4-0)/n = 4/n# Find #x_i# And #x_i = a+iDeltax = 0+i4/n = (4i)/n# Find #f(x_i)# #f(x_i) = x_i/2 =1/2 * x_i# # = 1/2 * (4i)/n# # = (2i)/n# Find and simplify #sum_(i=1)^n f(x_i)Deltax # in order to evaluate the sums. #sum_(i=1)^n f(x_i)Deltat = sum_(i=1)^n ( (2i)/n) 4/n# # = sum_(i=1)^n( (8i)/n^2)# # = 8/n^2 sum_(i=1)^n(i)# Evaluate the sums # = 8/n^2((n(n+1))/2)# (We used a summation formula for the sums in the previous step.) Rewrite before finding the limit #sum_(i=1)^n f(x_i)Deltax = 8/n^2((n(n+1))/2)# # = 4((n(n+1))/n^2))# Now we need to evaluate the limit as #nrarroo#. #lim_(nrarroo) ((n(n+1))/n^2) = lim_(nrarroo) (n/n*(n+1)/n) = 1# To finish the calculation, we have #int_0^4 x/2dx = lim_(nrarroo) ( 4((n(n+1))/n^2))# # = 4(1) = 4#

Isaiah Allen · Answer

4

#int_0^4x/2dx# #=[x^2/2xx1/2]_0^4# #=[x^2/4]_0^4# #=[x^2/4]^4-cancel([x^2/4]_0# #=4^2/4=16/4=4#

Aurora Alvarado · Answer

undefined To evaluate the definite integral ∫(x/2)dx from 0 to 4 using the limit definition, you follow these steps:

1. Write the integral as a limit of Riemann sums:
   ∫(x/2)dx = lim(n→∞) Σ[(f(xi)*Δx)] from i=1 to n

2. Determine the width of each subinterval, Δx:
   Δx = (b - a) / n
   where 'a' is the lower limit (0), 'b' is the upper limit (4), and 'n' is the number of subintervals.

3. Choose sample points within each subinterval:
   xi = a + iΔx
   where 'i' ranges from 0 to n.

4. Evaluate the function at each sample point:
   f(xi) = xi / 2

5. Plug these values into the Riemann sum formula and simplify.

6. Take the limit as the number of subintervals approaches infinity to find the definite integral.

In this case:
a = 0, b = 4
Δx = (4 - 0) / n = 4/n
xi = 0 + i(4/n) = 4i/n

Now, we plug these values into the Riemann sum formula:
lim(n→∞) Σ[(xi/2)*(4/n)]
= lim(n→∞) Σ[(4i/n)*(4/n)]
= lim(n→∞) Σ[(16i/n^2)]
= lim(n→∞) (16/n^2) * Σ[i]

Using the formula for the sum of the first 'n' positive integers, Σ[i] = n(n + 1)/2:
= lim(n→∞) (16/n^2) * [n(n + 1)/2]

After simplifying and taking the limit as n approaches infinity, you'll get the result of the definite integral.