Respuesta :
Answer and Step-by-step explanation: The Trapezoidal and Simpson's Rules are method to approximate a definite integral.
Trapezoidal Rule evaluates the area under the curve (definition of integral) by dividing the total area into trapezoids.
The formula to calculate is given by:
[tex]\int\limits^a_b {f(x)} \, dx = \frac{b-a}{2n}[f(x_{0})+2f(x_{1})+2f(x_{2})+...+2f(x_{n-1})+f(x_{n})][/tex]
The definite integral will be:
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{9-4}{2.8}[2+2.\sqrt{5} +2.\sqrt{6} +2.\sqrt{7}+2.\sqrt{8}+3][/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{5}{16}[25.3193][/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = 7.9122[/tex]
Simpson's Rule divides the area under the curve into an even interval number of subintervals, each with equal width.
The formula to calculate is:
[tex]\int\limits^a_b {f(x)} \, dx = \frac{b-a}{3n}[f(x_{0})+4f(x_{1})+2f(x_{2})+...+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})][/tex]
The definite integral will be:
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{9-4}{3.8}[2+4.\sqrt{5} +2.\sqrt{6} +4.\sqrt{7} +4\sqrt{8} +3][/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{5}{24}[40.7398][/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = 8.4875[/tex]
Calculating the definite integral by using the Fundamental Theorem of Calculus:
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \int\limits^9_4 {x^{\frac{1}{2} }} \, dx[/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{2.\sqrt[]{x^{3}} }{3}[/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{2.\sqrt[]{9^{3}} }{3}-\frac{2.\sqrt[]{4^{3}} }{3}[/tex]
[tex]\int\limits^9_4 {\sqrt{x} } \, dx = 12.6667[/tex]
Comparing results, note that Simpson's Rule is closer to the exact value, i.e., gives better approximation to the exactly value calculated by the fundamental theorem.
