Respuesta :
Answer:
Given definite integral as a limit of Riemann sums is:
[tex] \lim_{n \to \infty} \sum^{n} _{i=1}3[\frac{9}{n^{3}}i^{3}+\frac{36}{n^{2}}i^{2}+\frac{97}{2n}i+22][/tex]
Step-by-step explanation:
Given definite integral is:
[tex]\int\limits^7_4 {\frac{x}{2}+x^{3}} \, dx \\f(x)=\frac{x}{2}+x^{3}---(1)\\\Delta x=\frac{b-a}{n}\\\\\Delta x=\frac{7-4}{n}=\frac{3}{n}\\\\x_{i}=a+\Delta xi\\a= Lower Limit=4\\\implies x_{i}=4+\frac{3}{n}i---(2)\\\\then\\f(x_{i})=\frac{x_{i}}{2}+x_{i}^{3}[/tex]
Substituting (2) in above
[tex]f(x_{i})=\frac{1}{2}(4+\frac{3}{n}i)+(4+\frac{3}{n}i)^{3}\\\\f(x_{i})=(2+\frac{3}{2n}i)+(64+\frac{27}{n^{3}}i^{3}+3(16)\frac{3}{n}i+3(4)\frac{9}{n^{2}}i^{2})\\\\f(x_{i})=\frac{27}{n^{3}}i^{3}+\frac{108}{n^{2}}i^{2}+\frac{3}{2n}i+\frac{144}{n}i+66\\\\f(x_{i})=\frac{27}{n^{3}}i^{3}+\frac{108}{n^{2}}i^{2}+\frac{291}{2n}i+66\\\\f(x_{i})=3[\frac{9}{n^{3}}i^{3}+\frac{36}{n^{2}}i^{2}+\frac{97}{2n}i+22][/tex]
Riemann sum is:
[tex]= \lim_{n \to \infty} \sum^{n} _{i=1}3[\frac{9}{n^{3}}i^{3}+\frac{36}{n^{2}}i^{2}+\frac{97}{2n}i+22][/tex]
The integral is equivalent to the following Riemann sum: [tex]\int_{4}^{7} \left(\frac{x}{2} + \frac{x^{3}}{3} \right)\,dx = \lim_{n \to \infty} \Sigma\limits_{i=1}^{n} \left[\frac{4+i\cdot \Delta{x}}{2} + \frac{(4+i\cdot \Delta{x})^{3}}{3}\right][/tex]
A Riemann sum consist in a discrete approximation to the area below the curve by using sums of rectangles, whose accuracy depends on the number of rectangles used. An integral can be seen as a Riemann sum with infinite number of rectangles. If the Riemann sum uses right endpoints, then the integral is described by the following Riemann sum:
[tex]\int_{4}^{7} \left(\frac{x}{2} + \frac{x^{3}}{3} \right)\,dx = \lim_{n \to \infty} \Sigma\limits_{i=1}^{n} \left[\frac{4+i\cdot \Delta{x}}{2} + \frac{(4+i\cdot \Delta{x})^{3}}{3}\right][/tex], where [tex]\Delta x =\frac{3}{n}[/tex]. (1)
The integral is equivalent to the following Riemann sum: [tex]\int_{4}^{7} \left(\frac{x}{2} + \frac{x^{3}}{3} \right)\,dx = \lim_{n \to \infty} \Sigma\limits_{i=1}^{n} \left[\frac{4+i\cdot \Delta{x}}{2} + \frac{(4+i\cdot \Delta{x})^{3}}{3}\right][/tex]
Nota - The statement presents typographical mistakes, the correct form is presented below:
Express the integral as a limit of Riemann sums. Do not evaluate the limit. (Use the right endpoints of each subinterval as your sample points.)
[tex]\int \limits_{4}^{7}\left(\frac{x}{2} + x^{3} \right)\,dx[/tex]
We kindly invite to check this question on Riemann sums: https://brainly.com/question/23960718
