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Numerade Educator

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Problem 33 Medium Difficulty

The graph of $ f $ is shown. Evaluate each integral by interpreting it in terms of areas.

(a) $ \displaystyle \int^2_0 f(x) \, dx $
(b) $ \displaystyle \int^5_0 f(x) \, dx $
(c) $ \displaystyle \int^7_5 f(x) \, dx $
(d) $ \displaystyle \int^9_0 f(x) \, dx $

Answer

(a) 4
(b) 10
(c) -3
(d) 2

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Video Transcript

with. Now knowing what your graph is, the best I can do is give you an idea of how to solve this. You need to remember that integration is just Area. So if I'm asked to find the interval from 0-2 on my graph market off and look at what it is. In this case it's a triangle. Remember the area of the triangle is base times height divided by two. So the bases to the height is too and two times two divided by two or one half the base times height is to. So this area here is two units. Then the next thing you're asked Is the integral from 0 to 5. So now I'm gonna come over to five now I already know it's too so I've already got that part done too. And now I look and see, okay well this is a rectangle in the area of a rectangle is base so it's 123 units times the height. It's also two units long, so two plus six. So this would be six And so the area from 0 to 5 is eight. And then you're asked to find The interval from 5 to 7. So looking at my graph, make them a little dash here. I see I have a trapezoid and a trapezoid. I need to Add the parent length of the parallel size together. So this goes up to white four. So this sites four. So I'm going to add the parallel sides together. Multiply that by This width here which is two units and then take a half, So 1/2 times six times 2. So this particular in the role, This part of it would be six units long. So from 5 to 7 it would be six. And then finally you're asked to find the area From zero all the way tonight. So I already know I have two plus six plus another six. Sorry you have 14 units. Now I look at this last little portion and see it's also a triangle basis to height is for so base times height, 1/2 base times site is going to give me four. It's going to add another four. So the total area would be 18 units. So you need to break your picture up into geometric figures. Um if it's a circle you'll use pi r squared. But if it's a triangle or rectangle or a trapezoid, you can use your geometry formulas.