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Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid x \ge 0, 0 \le y \le xe^{-x} \} $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

01:56

Sketch the region and find…

01:40

05:02

06:53

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02:29

the problem is sketch the region and find its area. Look at this graph we need to find It's the roof area off. It's a shady depart this part we can use and grow from the road to sanity of it X temps in connected last e x to denounce area on this part. But this improper, integral. But that mission, this is equal to the limit. It goes to infinity into go from zero to a ex Thames into make two blacks. Yes, we computed this definite integral first for this death it into zero. We used my third off integration, My parse. This is equal chewed into girl from there to a thanks. The conectiv in tow make you x use the my third wind charisma. Pars, this is Echo two ex rams. Negative, you two. Next, Lex. From zero to a minus into door from zero to a negative to make you X. Yes, thiss is a culture plugging a zero to this function. Is that control connective? A ham too, Next to fade minus zero. Minus the class. My last month in the class into your off into Mickey Lax from there to a Zico connective eh into negative, eh? Maya's Tio? Maybe to us from zero. Since this is Echo Two negative, eh? Into the night away minus into negative, eh? Minus into make two zero minus one. There's someone I want a goes toe infinity! It's what it goes to zero. This part also goes to D'Oh. What's this one? You can use the limit. It goes to zero. This negative a over into a Now it goes toe infinity and then use a bean Has rules. This is culture. Damn It makes you want over you too, eh? It goes to infinity. This is gusto. Here. His answer is is zero minus Make to want. So this is you. Go to one. His area of this region is equal to one.

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