Question

let
f(x) = 80 + 4x − 4x2
and
g(x) = 4x2 − 20x.
Sketch the region enclosed by the graphs of
f(x)
and
g(x)
and compute its area.

Answer 1

did you graph them yet? when you do, locate their points of intersection. these will be your limits of integration

Answer 2

i got 457.333 but it's not right.

Answer 3

What did you get for the points of intersection ?

Answer 4

i took 80+4x-4x^2=4x^2-20x and then equalled it to zero

Answer 5

0=8x^2-24x-80

Answer 6

0=8(x^2-3x-10)
0=8(x+2)(x-5)
x=-2,5

Answer 7

perfect, these will be the limits of integration

next we just determine which function is larger on that interval

Answer 8

\[\int\limits_{-2}^{5}(80+24x-8x^2)dx\]

Answer 9

what do i do now though?

Answer 10

integrate it

it becomes

80x + 12x^2 - 8/3 x^3

Answer 11

(80(5)+12(5)^2-8/3(5)^3)-(80(-2)+12(-2)^2-8/3(-2)^3)=457.333

Answer 12

thats it. maybe if you dont want to estimate just use

1372/3

Answer 13

i just entered it onto my online homework and that was the problem. thank you so much.

Answer 14

yeah those things are picky :\

Answer 15

looks like they are midpoints, each with a base of 1

so calculate the function's value at those midpoints and determine the area that way

Answer 16

just plug the -1,-2,-3,1,2,3 into the x in the equation and add them up?

Answer 17

well no

the base for all rectangles is 1 but their heights are the midpoints evaluated at the function

Answer 18

so your x values would be -2.5, -1.5, -0.5, 0.5, 1.5, 2.5