Estimate the area under the graph of f(x) = 5 + 2x2 from x = −1 to x = 2 using three rectangles and right endpoints.

Question

Estimate the area under the graph of f(x) = 5 + 2x² from x = −1 to x = 2

 using three rectangles and right endpoints.

R₃=_______

Then improve your estimate by using six rectangles.

R₆=______

Answer 1

x=0, 1,2

f(0) = 5

f(1) = 7

f(2) = 13

deltaX=1

area is 1*(5+7+13) = 25

now for 6 rectangles:

x = -1/2, 0, 1/2, 1, 3/2, 2

f(-1/2) = 11/2

f(0) = 5

f(1/2) = 11/2

f(1) = 7

f(3/2) = 19/2

f(2) = 13

deltaX = 1/2

area is 1/2* ( 11/2+5+11/2+7+19/2+13)

= 1/2 * ( 11+10+11+14+19+26)/2

= 91/4

= 22.75

the real area is 5x + (2/3)x^3 , x = -1..2

x-->2 : 5(2) + (2/3)(2^3) = 10 + 16/3 = 46/3

x-->-1: 5(-1) + (2/3)(-1)^3 = -5 + -2/3 = -17/3

subtracting: (46 - -17)/3 = 63/3 = 21