Problem

Determine whether the statement is true or false.…

01:07

Problem 18 Medium Difficulty

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=3 x-3 ;[a, x]=[2, x]
$$

Answer

$A(x)=\frac{3}{2} x^{2}-3 x$

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Chapter 5

INTEGRATION

Section 1

An Overview of the Area Problem

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Video Transcript

So in this problem, we're working with the function F of X is equal to three X minus three and we're working on the interval from two to some X value. So in order to calculate the area under the curve, what I'm going to do here is first draw the picture of the curve and then figure out what shape geometrically will help us to calculate the area. In this particular case, I have a why intercepted negative three and a slope of three. So up. 3/1 up, three over. One up, 3/1 up, 3/1. Here's my line. I know that my shape will start at two and go somewhere in this direction because it has to go to is the lower bound and it's gonna end it some X value. Well, the shape that I'm going to get here is going to be a trap. Is oy in order? Calculate the area of a trapezoid. I'm going to do one half the height, times the sum of the basis. In this case, this would be based one. This would be based to, and this would be my height from my trap is oId here is just sideways. So my height value here is gonna end it X and start at two. So if I take one half, if I take X minus two, that will give me the height of my trap site based one will have an output value of three. We can see that right here based to is gonna have some output value Appears that's gonna lie on the line at whatever the X value is which will be determined by the function, which is three X minus three. Cleaning up this function will give us an equation that will help us to calculate the area under the curve at some ending value backs. So I have one half times X minus two and then three and negative three will cancel times three x. So this is the formula. I can't simplify this. I could multiply it out. But this can't really be simplified much more. This is the formula that will allow us to calculate the area under the curve on the interval from two to some X value

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