MAT 103 Assignment Problem 5.2

MAT 103 Assignment Problem 5.2 | Borough of Manhattan Community College

5.2 Area

Question 1

(1 point) Find the sum.

Question 2

(1 point) Find the sum.

Question 3

(1 point) Find a formula for the sum of nn terms. Use the formula to find the limit as n→∞n→∞

limn→∞∑i=0n(1+2in)22n

Ans 3

Question 4

1 point) Find a formula for the sum of nn terms. Use the formula to find the limit as n→∞n→∞

limn→∞∑i=0n(2+3in)33n

  Question 5

(1 point) The rectangles in the graph below illustrate a left endpoint upper sum for f(x)=13xf(x)=13x on the interval [2,6][2,6].
The value of this left endpoint upper sum is 15.4251 , and this upper sum is an        ?  overestimate of  equal to  underestimate of  there is ambiguity  the area of the region enclosed by y=f(x)y=f(x), the x-axis, and the vertical lines x = 2 and x = 6.

Question 6

(1 point)

Question 7

(1 point)

Question 8

(1 point)

Consider the function f(x)=e−x2,−2≤x≤2.

Question 9

(1 point) Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval.

y=3x−2,[2,4]y=3x−2,[2,4]

Question 10

(1 point) (A) Estimate the area under the graph of

f(x)=2x3+3f(x)=2x3+3

from x=−1x=−1 to x=5x=5, first using 6 approximating rectangles and right endpoints, and then improving your estimate using 12 approximating rectangles and right endpoints.

Question 11

(1 point) Definition: The area AA of the region SS that lies under the graph of the continuous function ff is the limit of the sum of the areas of approximating rectangles

A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx].A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx].

(a)   Use the above definition to determine which of the following expressions represents the area under the graph of f(x)=x3f(x)=x3 from x=0x=0 to x=1x=1.

Ans 11

A. limn→∞∑i=1n(in)35nlimn→∞∑i=1n(in)35n
B. limn→∞∑i=1n(in)31nlimn→∞∑i=1n(in)31n (correct)
C. limn→∞∑i=1n(in)1nlimn→∞∑i=1n(in)1n
D. limn→∞∑i=1n(in)5nlimn→∞∑i=1n(in)5n


(b) Evaluate the limit that is the correct answer to part (a). You may find the following formula for the sum of cubes helpful:

13+23+33+⋯+n3=∑i=1ni3=(n(n+1)2)2.13+23+33+⋯+n3=∑i=1ni3=(n(n+1)2)2.

The value of the limit is 14​ .

Question 12

Definition: The AREA A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles

A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+...+f(xn)Δx]

Ans 12

(a) Use the above definition to determine which of the following expressions represents the area under the graph of f(x)=x5f(x)=x5 from xx = 0 to xx = 2.

A. limn→∞64n6∑i=1nilimn→∞64n6∑i=1ni
B. limn→∞64n6∑i=1ni5limn→∞64n6∑i=1ni5(correct)
C. limn→∞1n6∑i=1ni5limn→∞1n6∑i=1ni5
D. limn→∞64n∑i=1ni5limn→∞64n∑i=1ni5

Question 13

(1 point) (A) Estimate the area under the graph of

f(x)=36−x2f(x)=36−x2

from x=0x=0 to x=6x=6 using 6 approximating rectangles and right endpoints.

Ans13

Estimate = 125

(B) Repeat part (A) using left endpoints.

Estimate = 161
(C) Repeat part (A) using midpoints.

Estimate = 144.5

Question 14

(1 point) Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval.

y=3x2+1,[2,4]y=3x2+1,[2,4]

Ans 14

58

Question 15

(1 point) Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval.

y=2x3−x2,[4,5]y=2x3−x2,[4,5]

Ans 15

\frac{985}{6}