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Find $ \displaystyle \int^5_0 f(x) \, dx $ if

$ f(x) = \left\{ \begin{array}{ll} 3 & \mbox{if $ x < 3 $}\\ x & \mbox{if $ x \ge 3 $} \end{array} \right.$

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01:18

Frank Lin

00:36

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Oregon State University

Harvey Mudd College

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.

00:34

Find $$\int_{0}^{5} f(x) d…

01:08

Find $\int_{0}^{5} f(x) d …

01:00

Given $\int_{-.5}^{3} f(x)…

00:54

Express as one integral.

00:59

$$\text { Given } \int_{0}…

00:53

Suppose that $\int_{3}^{5}…

00:32

$$\begin{array}{l}{\te…

00:47

Find $\int_{1}^{5} f(x) d …

we have the integral from 0 to 5 and three is the height from 0 to 3. And then why equals X is the height from three and beyond. Then we want to find the area under the two curves. So the area under this rectangle here would just be three times three or nine and the area under this blue. Why, it was X line there. It's going to be what we could do this with a trapezoid, Um, and so that would end up being a base of two. Actually, you know, let's just do a rectangle on the triangle instead. So that would be two times three. So that would be an area of six there, and then that would be a height of two with of 22 times 24 divided by two. So that would be in the area of two up top. So just so you can see both of those, that's to that six. And that's of course, our first nine. So the total area under the curve there is going to be 17 9 plus six plus two. Therefore, that's what the Senate girl evaluates dio

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