The area under the graph of f(x) = 2 + 4x^2 from x = -1 to x = 2 has to be determined using different methods.

a) Using three rectangles and right endpoints.

The area under the graph is approximated by 3 rectangles.

The width of each of the rectangles is (2 - (-1))/3 = 3/3 = 1.

Length of the three rectangles starting from right is,

R1 = f(2) = 18

R2 = f(1) = 6

R3 = f(0) = 2

The sum of their area is 1*18 + 1*6 + 1*2 = 26.

b) Using six rectangles and right endpoints,

The width of each of the rectangles is (2 - (-1))/6 = 3/6 = 1/2.

The length of each rectangle starting from the right is,

R1 = f(2) = 18

R2 = f(1.5) = 11

R3 = f(1) = 6

R4 = f(0.5) = 3

R5 = f(0) = 2

R6 = f(-0.5) = 3

The sum of their area is 0.5*18+0.5*11+0.5*6+0.5*3+0.5*2+0.5*3 = 21.5.

c) Using the left endpoints method, for 3 rectangles the width of each rectangle is equal to (2 - (-1))/3 = 1.

The length of the rectangles starting from the leftmost one is,

L1 = f(-1) = 6

L2 = f(0) = 2

L3 = f(1) = 6

Adding the area of these rectangles gives an...

(The entire section contains 2 answers and 570 words.)

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