Answer :
Answer: The answer is 152.25 sq units.
Step-by-step explanation: Given function to be integrated is
[tex]f(x)=x^3,~~x=2~~\textup{to}~~x=5.[/tex]
To find the area of the given curve from x = 2 to 5, first we need to integrate the function and we will put the boundary values and subtract the smallest from largest value.
The Riemann sum and the formula to find the area is given by
[tex]A=\int_{x=2}^{5}f(x)dx\\\\\Rightarrow A=\int_{2}^{5}x^3dx\\\\\Rightarrow A=[\dfrac{x^4}{4}]_2^5=\dfrac{1}{4}(625-16)=152.25[/tex]
Thus, the required area is 152.25 sq units.
The area under the graph of the function [tex]f\left( x \right) = {x^3}[/tex] from [tex]x = 2[/tex] to [tex]x = 5[/tex] is [tex]\boxed{152.25{\text{ unit}}{{\text{s}}^2}}.[/tex]
Further explanation:
Given:
The function is [tex]f\left( x \right) = {x^3}.[/tex]
The function is defined in the interval from [tex]x = 2[/tex] to [tex]x = 5.[/tex]
Explanation:
The given function is [tex]f\left( x \right) = {x^3}.[/tex]
Integrate the given function with respect x.
[tex]\begin{aligned}Area &= \int\limits_2^5 {f\left( x \right)dx} \\&= \int\limits_2^5 {{x^3}dx}\\&= \left[ {\frac{{{x^4}}}{4}} \right]_2^5\\&= \left( {\frac{{{5^4}}}{4} - \frac{{{2^4}}}{4}} \right)\\\end{aligned}[/tex]
Further solve the above equation to obtain the area under the curve,
[tex]\begin{aligned}{\text{Area}} &= \frac{{625}}{4} - \frac{{16}}{4}\\&= \frac{{625 - 16}}{4}\\&= 152.25\\\end{aligned}\\[/tex]
The area under the graph of the function [tex]f\left( x \right) = {x^3}[/tex] from [tex]x = 2[/tex] to [tex]x = 5[/tex] is [tex]\boxed{152.25{\text{ unit}}{{\text{s}}^2}}.[/tex]
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Riemann function
Keywords: Riemann, sum, area, graph function, Riemann sum, area under the curve, function, [tex]f(x) -= x3, x = 2, x = 5.[/tex]
