Question

what is the area of the region enclosed by the graph of f(x)=x - 2x^2 and g(x) = -5x? i do not have a graphing calculator

Answer 1

\[f(x)=x-2x ^{2} \]
\[g(x)=-5x\]

Answer 2

you need to start out by setting both of those equations equal to each other. One is parabola the other is line so to close off an area using those equations would require the line to intersect with the parabola at 2 points, meaning your solution from setting them equal to each other should have 2 answers. next you will use those two points as the upper and lower boundaries and then you integrate

Answer 3

how do i figure out the limits of integration?

Answer 4

nevermind, sorry

Answer 5

are the limits 0 and 3?

Answer 6

you got it

Answer 7

how do i figure out the function i integrate?

Answer 8

essentially you are integrating both but the next step after you determine your limits of integration is to figure out what the top and bottom functions are, this step is obviously way easier if you have a graphing calc but you can also intuit it just by thinking through what these graphs have to look like and drawing a sketch. then you have to subtract the "top" function from the bottom because remember that integration is the whole area under the curve between your limits of integration, but you only want the part that is closed in.

Answer 9

then you integrate!

Answer 10

is the top curve x-2x^2 ?

Answer 11

yep, its usually pretty obvious once you look at the graph but you just gotta be careful during this if you have something like a sideways parabola or a circle in which case the same function can be on the top and bottom

Answer 12

um here is a visual representation of what i'm thinking. is it true that the integral of the x-2x^2 is the shaded area?