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Use Riemann sums and the limit to find the area.Find the area of the region bounded by $f(x)=x^{3}$ and the $x$ -axis, between (a) $x=1$ and $x=2,$ (b) $x=-1$ and $0,(\text { c) } x=-1 \text { and } 1,(\text { d) } x=-2 \text { and } 1$.

(a) $15 / 4$(b) $1 / 4$(c) $1 / 2$(d) $17 / 4$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Baylor University

Idaho State University

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

02:47

Consider the area bounded …

09:22

Use the method of Riemann …

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Write the area of each reg…

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02:29

Graph the following functi…

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Determine the area of the …

01:32

Okay, So in this problem, they actually give you the area of its equal to one half, which is different than some of the other problems. Because if we're looking for the area and mashed with the X axis, uh, we have to do the integral mhm. Uh, they do tell you that X equals zero, and they do tell you that x equal to be so we're doing the integral from zero to be of the function and give you which it looks like it's x cubed plus one. Mhm. Yeah. Yeah, I guess that's all I really need. DX um and so this is the equation for the area. So I don't know if it's obvious to you, but this is the equation for the area, and they tell you the area is one half which is set equal before messing with that equals one half. What I'm gonna do is take the left side and do the anti driven, which is adding one to the exponents and divided by a new X women or multiplied by their secrets. And that's going to be from zero to be. So I'm ignoring this equally one half for a second I'll use that in a second. Now plug in your bounds. So 1/4 B to the fourth plus V. Uh, and you have to also plug in zero in from both of us. But you're going to get two zeros, so it's almost always a time is equal to one half. Um and I'm from here. I think what you would probably need to do is use a graphing calculator where you set one equation. Oh, I don't think this respectable. Uh, let's be now. I guess I should clarify. This is I'm writing the left side of the equation, not writing those zeros. And the only thing is, with a graphing calculator, you probably need to rate X to the fourth plus X and then with the second equation, set that equal to one half, which is just a horizontal line. But examine these two and find the point of intersection. And when you find that point of intersection, that's your B value, because instead of the b you're doing so, whatever, the X answer is going to be the p value. Um, and the answer I came up with I liked around 2 a.m. three decimals. Negative zero point 51 eight. Supposed to circle and green. Yeah,

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