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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 $

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

01:15

Frank L.

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Campbell University

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University of Nottingham

Idaho State University

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(a) Graph $f(x)=x(x+2)(x-1…

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The figure shows the areas…

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to find out the area. For each of these, we could go ahead and break them up into squares and triangles. So the first one has two squares on the bottom. It's the area from X zero access to, and then we have a triangle line right above it, which has a base of two height of two. So that has an area of two. So that would give us a total of an area of four there for that first one. Then, from 0 to 5, we could go ahead and take the four that we already left off with, plus an additional three squares from 2 to 3 and then plus, then we have a triangle there from 3 to 5, and that has a height of three. A base of two one half base times height, basically six divided by two is three. So all of that area from 0 to 5 is 10. Then that's the positive area. So now we know we're going to start going downwards here, so we're have negative area from 5 to 7. That also is a triangle that has a width of two height of negative three and therefore two times negative three negative six divided by two that has an area of negative three things. And then lastly, from 0 to 9, we could go ahead and combined what we got from 0 to 5, which was 10 then from pick up from 5 to 7, which is negative. Three. And then lastly, we wanna take away the area from 7 to 9 since that's also negative. And let's go ahead and draw out what that is. That shape kind of looks like this from from five. Sorry from 7 to 9 and that is 1234 here negative and a negative one underneath that for a total of negative five. Therefore, when we combine all these together, that last one will have an area of positive too, because it's basically positive. 10 above the X axis Negative eight below the X axis. So combining those get positive too

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