Estimate integral from-5 to 7 of (-3x-1x^2)dx using a Riemann sum, midpoints, and n=3 subintervals.

Answer 1

-3x -x^2 = f(x)

F(x) = -3x^2/2 - x^3/3 + c

evaluated from -5 to 7

= F(7)-F(-5)

= -3(49)/2 - 343/3 -[-(25)/2 - 125/3] +c - c

=-147/2 -343/3 + 25/2 + 125/3

=-122/2 - 218/3

= -366/6 - 536/6

= - 902/6

= -151 1/3

do Reiman sums and you should get somewhere in the range of that same definite integral

3 subintervals mean (7--5)/3 = (7+5)/3 = 12/3 = 4 = the base of each rectangle, from -5 to -1, -1 to 3, and 3 to 7

midpoints are the average of the endpoints: -3, 1, and 5

heights of each rectangle are:

f(-3) = -3(-3) - (-3)^2 = 9-9=0

f(1) = -3(1) - (1)^2 = -3-1 = -4

f(5) = -3(5) -(5)^2 = -15-25 = -40

areas of the rectangles are sum of the bases times heights:

4(0) -4(4) -4(40) = -16-160 = -176