Evaluating integrals Evaluate the following integrals. ∫0^x...

Key Concepts

Double Integrals

Double integrals extend the concept of a single definite integral to functions of two variables, allowing the calculation of volumes under surfaces over a specified region in the plane. They are fundamental in multivariable calculus and are frequently computed as iterated integrals, where one integrates one variable at a time.

Iterated Integrals

Iterated integrals are a method for evaluating double or multiple integrals by performing integration sequentially with respect to one variable at a time. This approach typically involves integrating the inner integral while treating the other variables as constants, and then integrating the resulting expression with respect to the next variable.

Integration Limits and Region Specification

Understanding integration limits is essential in setting up and evaluating double integrals, as they define the region of integration in the plane. The limits determine how the inner and outer integrations are performed, and careful interpretation of these boundaries is necessary to accurately compute the integral over the desired region.

Transcript

0:00 Hello.

00:01 So today we're going to a double integral, and we're asked to evaluate that double integral.

00:06 So first, let's break it down into parts.

00:12 So we see that first we have to differentiate with respect to x, and this is also known not only by the dx that's closest, but also by the fact that we're evaluating at y.

00:25 And then we also, for our second integral, are going to be integrating with respect to y.

00:33 So doing the first integral, it's pretty straightforward.

00:36 Well, what's the integral at the x? that's x.

00:39 And we'll evaluate that from 0 to y cosine of y.

00:45 So plugging that in, we have y, cosine of y minus 0.

00:52 And then the next integral is from 0 to x divided 2, d y.

01:00 So now we're given a y -cosy.

01:03 And this is where we actually have to integrate by parts.

01:10 Integrate five parts.

01:15 And there's a special formula in a special way that do this.

01:18 So a lot of times people will use the notation u and v.

01:23 So let's take u equal to y and du equal to dy.

01:32 But then we also have to take dv and v, where v or sorry dv is our second part.

01:42 So that would be our cosine of y and so the integration of v well what's the integral of cosine that's sign of y and then don't forget your d y there so now that you break down to the part so we have our u part which is our y and we have our dv part which equals cos y and then that brings us to the standardized formula for integration by parts so when you have a ub type scenario which we do we have our y and our pose of y.

02:19 So to integrate this, the integral is u times v minus the integral of v du...

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