Sketch the region enclosed by the given curves and find its area. y=e^x, y=x e^x, x=0

Name: Problem 9, Chapter 6: Biocalculus Calculus for the Life Sciences
Uploaded: 2020-04-05T10:23:13-08:00
Duration: 10 min 21 s
Channel: Sinisa Stura
Description: Sketch the region enclosed by the given curves and find its area. y=e^x, y=x e^x, x=0

step 1 Hello, we are given three curves. Y equals E to the power of X, Y equals X times, E to the power

Sketch the region enclosed by the given curves and find its...

Key Concepts

Graphical Analysis

Sketching the graphs of the functions provides a visual understanding of the region of interest. It helps in identifying which function is above or below over the interval, assists in setting the correct limits of integration, and ensures the accurate formulation of the integral used to compute the area.

Definite Integration

Definite integration is used to calculate the area under a curve within specified limits. In this context, it involves constructing an integral where the integrand is the difference between the functions that define the upper and lower boundaries, evaluated between the determined intersection points.

Area Between Curves

This concept involves computing the area enclosed between two or more curves by setting up a definite integral. It requires determining which function forms the upper and lower boundaries over a certain interval and then integrating the difference between them. It is a fundamental application of definite integration in calculus.

Intersection Points

Finding the points where curves intersect is crucial in determining the limits of integration for the area computation. These points are the solutions to the equation obtained by setting the functions equal to each other, and they define the region over which the integration is performed.

Transcript

00:02 Hello, we are given three curves.

00:04 Y equals e to the power of x, y equals x times, e to the power of x, and the third curve is x equal x equals 0.

00:18 We are tasked to find the area enclosed by these three curves.

00:26 So we spiral by sketching this by hand.

00:30 Okay, start with this curve, this is the easiest curve.

00:36 This is the easiest curve.

00:36 This is a little.

00:36 The y x.

00:43 Let's call this curve give it a function that y equals f of x equal e to the power of x.

00:53 We should know how to sketch this curve because we know it passes through points e to the power of 0 is 1.

01:00 E to the power of 1 is e so we have these two points here and e is around 2 .7 there and our curve.

01:17 Exponential curve passing curve one something like.

01:23 Okay, the next curve is the y equals function of x.

01:36 0, this is 0, so the curve 1, x equals 1, it's 1 times d to the power 1, equal, go through 1, e the same thing that we have here.

01:50 And if we take v minus 1, v minus 1 times e to the power so, e minus d to be somewhere here.

02:08 This curve will look something like this.

02:14 Okay, the area, the area we're interested in is this area here.

02:22 The area between the curves and geogic.

02:30 Ah, in this sketch, we see that x is between 0 and 1, because the intersection point, the intersection point between f of x and gioex is here.

02:45 And the y axis is that third curve so one three points determined region here to see this area more clearly we may use our friendly online free calculator at desmos .com and when we enter the we see the red graph is the e to the power of x the blue graph is the other function xxxxxx and this is the y -axis this is the area this is the area we want one piece of advice to check your answer i mean you can use desmos you can use desmos to find out the answer to this whole question what is the area in a moment okay the tool that we will use is this the area of a region that is bounded by a function f of x above whose graph is above the graph of the function g of x the x is the integral with bounds of our interval of x the interval we are talking about is this interval here so for x belonging to the interval 0 to 1 we're looking what is the area of the region bounded by this function above, f of x.

04:47 And from below the area is bounded by g of x.

04:52 All right...

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