The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles: A = lim R₂ = lim (ƒ (x1) Ax + f (x2) Ax+ ... + f (x) Ax) 11-00 72-00 (a) Use this definition to find an expression for the area under the curve y = x³ from 0 to 1 as a limit. F(;)Ki lim 812 (-)- Σ) i=1 F,()K= lim 72-0 lim 11+00 Σ i=0 n ( 3 n n (b) Use the following formula for the sum of the cubes of the first integers to evaluate the limit in part (a). 13+23+... 'n (n+1)` ²
The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles: A = lim R₂ = lim (ƒ (x1) Ax + f (x2) Ax+ ... + f (x) Ax) 11-00 72-00 (a) Use this definition to find an expression for the area under the curve y = x³ from 0 to 1 as a limit. F(;)Ki lim 812 (-)- Σ) i=1 F,()K= lim 72-0 lim 11+00 Σ i=0 n ( 3 n n (b) Use the following formula for the sum of the cubes of the first integers to evaluate the limit in part (a). 13+23+... 'n (n+1)` ²
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.CR: Chapter 7 Review
Problem 54CR
Question
![The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles:
A = lim R₂ = lim (ƒ (x1) Ax + f (x2) Ax+ ... + f (x) Ax)
11-00
72-00
(a) Use this definition to find an expression for the area under the curve y = x³ from 0 to 1 as a limit.
F(;)Ki
lim
812
(-)-
Σ)
i=1
F,()K=
lim
72-0
lim
11+00
Σ
i=0
n
(
3
n
n
(b) Use the following formula for the sum of the cubes of the first integers to evaluate the limit in part (a).
13+23+...
'n (n+1)`
²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d3f739d-bc0c-464a-9c46-60dcfdea37ca%2F2f2cb70f-2c80-495f-a473-4a86f9867e5e%2F8qpzwx8_processed.png&w=3840&q=75)
Transcribed Image Text:The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles:
A = lim R₂ = lim (ƒ (x1) Ax + f (x2) Ax+ ... + f (x) Ax)
11-00
72-00
(a) Use this definition to find an expression for the area under the curve y = x³ from 0 to 1 as a limit.
F(;)Ki
lim
812
(-)-
Σ)
i=1
F,()K=
lim
72-0
lim
11+00
Σ
i=0
n
(
3
n
n
(b) Use the following formula for the sum of the cubes of the first integers to evaluate the limit in part (a).
13+23+...
'n (n+1)`
²
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