Math  /  Calculus

Question14(2x3)dx\int_{1}^{4}\left(2-\frac{x}{3}\right) d x using Diemann Sum

Studdy Solution

STEP 1

1. We are asked to approximate the integral 14(2x3)dx\int_{1}^{4}\left(2-\frac{x}{3}\right) dx using a Riemann sum.
2. We will use a Riemann sum with nn subintervals.
3. We will use the right endpoint of each subinterval for the Riemann sum.

STEP 2

1. Determine the width of each subinterval.
2. Find the right endpoint of each subinterval.
3. Evaluate the function at each right endpoint.
4. Construct the Riemann sum.
5. Take the limit as the number of subintervals approaches infinity.

STEP 3

Determine the width of each subinterval:
The interval [1,4][1, 4] is divided into nn subintervals, so the width Δx\Delta x is:
Δx=41n=3n\Delta x = \frac{4 - 1}{n} = \frac{3}{n}

STEP 4

Find the right endpoint of each subinterval:
The right endpoint of the ii-th subinterval is:
xi=1+iΔx=1+i3nx_i = 1 + i \cdot \Delta x = 1 + i \cdot \frac{3}{n}
for i=1,2,,ni = 1, 2, \ldots, n.

STEP 5

Evaluate the function at each right endpoint:
The function evaluated at the right endpoint xix_i is:
f(xi)=2xi3=21+i3n3f(x_i) = 2 - \frac{x_i}{3} = 2 - \frac{1 + i \cdot \frac{3}{n}}{3}
Simplify:
f(xi)=213in=53inf(x_i) = 2 - \frac{1}{3} - \frac{i}{n} = \frac{5}{3} - \frac{i}{n}

STEP 6

Construct the Riemann sum:
The Riemann sum RnR_n is given by:
Rn=i=1nf(xi)Δx=i=1n(53in)3nR_n = \sum_{i=1}^{n} f(x_i) \cdot \Delta x = \sum_{i=1}^{n} \left(\frac{5}{3} - \frac{i}{n}\right) \cdot \frac{3}{n}
Simplify:
Rn=i=1n(5n3in2)R_n = \sum_{i=1}^{n} \left(\frac{5}{n} - \frac{3i}{n^2}\right)
Rn=5ni=1n13n2i=1niR_n = \frac{5}{n} \sum_{i=1}^{n} 1 - \frac{3}{n^2} \sum_{i=1}^{n} i

STEP 7

Take the limit as nn approaches infinity:
Evaluate the sums:
i=1n1=n\sum_{i=1}^{n} 1 = n
i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
Substitute these into the Riemann sum:
Rn=5nn3n2n(n+1)2R_n = \frac{5}{n} \cdot n - \frac{3}{n^2} \cdot \frac{n(n+1)}{2}
Simplify:
Rn=53(n+1)2nR_n = 5 - \frac{3(n+1)}{2n}
Take the limit as nn \to \infty:
limnRn=532=10232=72\lim_{n \to \infty} R_n = 5 - \frac{3}{2} = \frac{10}{2} - \frac{3}{2} = \frac{7}{2}
The value of the integral is:
72\boxed{\frac{7}{2}}

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