Question

find the area of the region bounded by the parabolas f(x)=2x^2, g(x)=3x^2+5 and the line y=8.

Also find dy/dx given that y=1+x^2e^y

Answer 1

It helps to have a rough sketch of what the graphs look like in your mind.

Answer 2

Where did you get that, @agent0smith ?
Nice stuff :D

Answer 3

yeah but how do i actually calculate it without using graphing software?

Answer 4

@terenzreignz Google graph: https://www.google.com/search?btnG=1&pws=0&q=2x%5E2%2C+3x%5E2%2B5%2C+y%3D8

Then i just zoomed in, pasted in paint and made it funkayyyy.

@candacemerriman You don't need graphing software, that was just to show you what it looks like... it'll help set up the integral.

Answer 5

You'll want to break it up, like in this pic. Use the integral of (top function - bottom function).

For region 1, the top function is y=8, bottom function is 2x^2. (find your limits by finding the intersections of those two lines)

For region 2, top function is 3x^2+5, bottom function is 2x^2. (find your limits by finding the intersections of those two lines)

Then add them together, and double it, since it's symmetrical about the y axis, and because why work harder than you need to.

Answer 6

\[\Large Area = 2 \times \left[ \int\limits_{-2}^{-1} (8 - 2x^2) dx + \int\limits_{-1}^{0} (3x^2+5 - 2x^2) dx\right]\]

Answer 7

Thank you @agent0smith