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Given that $ \displaystyle \int^{\pi}_0 \sin ^4x \, dx = \frac{3}{8} \pi $, what is $ \displaystyle \int^0_{\pi} \sin ^4\theta \, d\theta $?

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00:43

Frank Lin

00:33

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

University of Michigan - Ann Arbor

University of Nottingham

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.

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Evaluate the following int…

given that the integral of signed to the fourth of X from zero to pi is 3/8 pie. We want to find out the exact same integral, of course, with a different variable with the limits flipped. And this is taking advantage of a property. Yeah, you'll see you over and over again and we'll jump right into it. It's just going to flip the area under the curve to become a negative when you flip the limits of integration. One way to illustrate this is if we looked at the Delta acts, for instance, but this first problem, it would end up being B minus a zero pi minus zero over and And then we could look at the turn, for instance as well. So that would end up really just simplifying too high. I over end now if we did that exact same thing with the next one, and we looked at X here we end up getting zero minus pi over and and then the I ith turned their eventually. But just in terms of the widths themselves, not ignoring even the I ith term. This one is positive. This one is negative. So that is just one demonstration of why needs values end up flipping. And when you flip the limits of integration, then it changes your answer to Ah, negative, whatever it waas.

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