Use the Definition to find an expression for the area under the graph of fas a limit. Do not evaluate the limit. Use the Definition to find an expression for the area under the graph of fas a limit. Do...
Use the Definition to find an expression for the area under the graph of fas a limit. Do not evaluate the limit.
Use the Definition to find an expression for the area under the graph of fas a limit. Do not evaluate the limit.
lim n---> infinity Sigma, with i = 1 and n on top
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write5,349 answers
starTop subjects are Math, Science, and Business
You need to evaluate the definite integral of the given funtion to find the area under its graph such that:
`int_1^n (x^2+ sqrt(1+2x)) dx `
You need to use the property of linearity of integrals such that:
`int_1^n (x^2+ sqrt(1+2x)) dx = int_1^n (x^2)dx+ int_1^n sqrt(1+2x)) dx `
`int_1^n (x^2+ sqrt(1+2x)) dx = x^3/3|_1^n + int_1^n sqrt(1+2x)) dx `
You need to use the following substitution to solve `int_1^n sqrt(1+2x)) dx` such that:
`2x+1 = t => 2dx = dt`
`int sqrt(1+2x)dx = int sqrt t dt/2 `
Converting the square root into a power yields:
`int sqrt t dt/2 = int t^(1/2) dt/2 = (1/2)(t^(1/2+1))/(1/2+1) + c`
`int sqrt t dt/2 = (t^(3/2))/3 + c => int_1^n sqrt(1+2x)dx = (1+2x)^(3/2)/3|_1^n`
`int_1^n (x^2+ sqrt(1+2x)) dx = (n^3 + (1+2n)^(3/2) - 1^3 - (1+2)^(3/2))/3`
`int_1^n (x^2+ sqrt(1+2x)) dx = (n^3 + (1+2n)^(3/2) - 1 - 3sqrt3)/3`
Hence, evaluating the area under the graph, between the limits 1 and n, yields `int_1^n (x^2+ sqrt(1+2x)) dx = (n^3 + (1+2n)^(3/2) - 1 - 3sqrt3)/3.`
Related Questions
- Use the Definition to find an expression for the area under the graph of f as a limit. Do not...
- 1 Educator Answer
- (a) Use Definition 2 to find an expression for the area under the curve y= (x^3) from 0 to 1 as a...
- 1 Educator Answer
- Use the definition to find an expression for the area under the graph of f as a limit. Do not...
- 1 Educator Answer
- Use this definition to find an expression for the area under the graph of f as a limit. Do not...
- 1 Educator Answer
- Use this definition to find an expression for the area under the graph of fas a limit. Do not...
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.