Problem

Graph each function over the specified interval. …

02:21

Problem 16 Easy 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.
$$
f(x)=3 x-3 ;[a, x]=[1, x]
$$

Answer

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

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Calculus 1 / AB

Calculus 2 / BC

Calculus Early Transcendentals

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 one to some X value. We're using our geometry to help us to find the area under the curve in this case because I don't know my ending points. Really? Just gonna give me a formula that will help us to calculate the area under the curve. So let's graph three X minus three means I have a y intercepted Negative. Three up, three over. One up, three over. One up, three over. One down, three back one and there's my line. I'm looking at the area under the curve, starting at one and going to some X value. I don't know where this X values gonna end, so I'm going to have a shape here. And if you look at this, the sheep here will end up looking like a triangle. So we're gonna use the area of a triangle which is one half the base times the height. Now the base unit here. Right, So we have one half times the base. The base will be from one toe X, So if I ended X and I subtract one unit that will be the length of the base. The heights will be the Y value at the particular X value, which is defined by my function, which is going to be three X minus three. In this particular case, this would be the area under the curve defined by the triangle. So I would use this formula once I knew what X was to calculate the area under the

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