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Problem 35 Easy Difficulty

Evaluate the integral by interpreting it in terms of areas.

$ \displaystyle \int^2_{-1} (1 - x) \, dx $

Answer

$\frac{3}{2}$

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Tommaso V.

April 18, 2021

The graph of f is shown below. Evaluate each integral by interpreting it in terms of areas. 1. ?20f(x)dx= 2 2. ?50f(x)dx=

TV

Tommaso V.

April 18, 2021

If ?81f(x)dx=18 and ?87f(x)dx=3.1, find ?71f(x)dx.

Video Transcript

All right, let's go ahead and answer the question. The integral from negative 1 to 2 off, one minus x dx. And then the question specifically asks you to evaluate using in terms off the geometrical approach. Okay, What does that mean? Basically, there are rules to integration that you can utilize to solve the problem analytically. But remember, earthy, integral can be conceptually thought, often area under the curve. So if you know how to calculate that area geometrically, you could also approach it that way. So let's analyze this on the X Y coordinate system. We are going to be able to graph this thing quite precisely. It's one of the first things that you learned in algebra. One minus X has three ey intercept that one. X intercept is also one because we know that the slope is negative. One. Okay, see if they can draw a nice street line. There you go. The limit of integration is from negative. Wanted to so negative one is right there. Two is right here. So what is the area under the curve? Well, it starts right here, and it ends over there so you can see that there are two wrecked ing two triangles right there and you can see this one is above the X axis. So it will be a positive area and this one is below the X axis. So it's going to be a negative area. You can also see right away that the red portion is larger than the blue portion. So the net area under the curve is going to be a positive number. So we immediately see that it's going to be a positive answer, okay? And these air triangles, which are very simple to calculate We've known this since maybe third or fourth grade. So that's calculated. The height here is going to be too. The distance from negative 1 to 1 is also too. So we know that the area of the red portion is for divided by two. Just took the blue region. You know, that wanted to is one the height here is one. So one divided by two is one half. And remember that this is a negative area because it's below the X axis. Okay, so the answer is going to be two minus one half, which is three over to. And that's how you answer this question.