Use the function y = 2x from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoint of each subinterval. a)What is the area of the first rectangle? b) What is the area of the second rectangle? c)What is the area of the ith rectangle?

Clifland 2021-02-13 Answered
Use the function y=2x from x=0 to x=1 and n equal subintervals with the function evaluated at the left-hand endpoint of each subinterval.
a)What is the area of the first rectangle?
b) What is the area of the second rectangle?
c)What is the area of the ith rectangle?
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Expert Answer

Maciej Morrow
Answered 2021-02-14 Author has 98 answers

Use the function y=2x from x=0 to x=1 and n equal sub-intervals with the function evaluated at the left-hand endpoint of each sub-interval.
Obtain the step size as follows.
Δ x=1  0n=1n.
Then the left-hand endpoint of the sub-intervals will be 0, 1n, 2n, , n  1n.
The function values at these endpoints will be 0, 2n, 4n, , 2(n  1)n
a)What is the area of the first rectangle?
Area of the first rectangle=length × width
=0 × 1n
=0
b)What is the area of the second rectangle?
Area of the second rectangle=length × width
=2n× 1n
=2n2
c)What is the area of the i th rectangle?
Area of the ith rectangle=length × width
=2(i  1)n × 1n  [ i=1, 2, , n]
=2(i  1)n2  [ i=1, 2, , n]
Hence, the area of the i th rectangle is 2(i  1)n2
 i=1, 2, , n

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