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Evaluate $ \displaystyle \int^1_1 \sqrt{1 + x^4}\, dx $.

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00:24

Frank Lin

00:20

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

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.

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$$\text { Evaluate } \…

01:06

Evaluate the definite inte…

06:23

Evaluate the integral.

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01:40

Evaluate the integral.…

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01:36

Evaluate.$$\int_{1}^{4…

02:09

All right, we are going to answer the question. What is the value of one integration from 1 to 1? The square root of one plus X to the fourth the X. Okay, I already know the answer. It only takes half a second to solve this thing. But why can I do that? It's actually based on this idea. The area between the curve and the X axis is written like this. Okay, so one of the first properties that you can see right away is that if the area under the curve it's on Lee from a T A f m x dx, it really doesn't matter what f of X is equal to. So if I try to grow, graph it. Let's say that I have a drawing like this. I started a and I end at a So what's the area under the curve off the line that has, with of zero. Well, I don't know what the height is going to be. I meant I don't know what the function is going to be, but ffx is gonna have ah, height f of A and Delta X is equal to zero. So the area it's just zero because it's f of eight times zero. Okay, So what is this equal to zero? What is this equal to? Well, from 1 to 1, the area under the curve, regardless of what this guy is, has to be zero. And that's how you answer this question.

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