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Sketch the region enclosed by the curves and find its area.$$y=x^{3}-4 x, y=0, x=0, x=2$$

$$4$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

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Okay, So this problem, the graph of the cubic looks, something like this, and it happens to be like right here is X equals zero Right here is X equals two. Um, so as faras, the equation goes, and they kind of gave you the values. Anyway, Do you only want this area? Um, now, the upper function is zero Weichel zero, and then you wanna subtract off the other function, which is the quantity of X cubed a minus four x and notice. I'm putting parentheses in here for one. You don't actually have to write this zero cause it's important to have that negative to make sure you have a positive answer. Um, but make sure you distribute that, but what you could dio, if you want another option is ah, just do the function and an absolute value. Your final answer by me to treat this is if this is actually the integral from 0 to 2 of negative x cubed plus four x dx. Okay, So the anna derivative, you add one to your exponents. So it's X to the fourth. I should put it putting equals in here, and then you have to divide by that new exponents. So divide by four. It's 1/4 same thing with this when you add one to your actual on it X squared and you divide by your new exponents of four. Divide by two is two, and that's from 0 to 2. Now it's really nice about this. Problem is, ah, when you plug in to to the fourth power plus two times two squared when you go to subtract off employees and zero for all these zero to the fourth power still +00 to the second power. Still zero. Um, so actually, this is a waste of time to write all that eso. Anyway, to the fourth power 248 16 divided by four is four two Squared is four times two is eight and Nega four plus eight is for your answer.

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