another question PLZ (1 Viewer)

[skool :(]

heyy again
plz can u answer this question too/ its also from fitzpatrick. ok here goes :
calculate the area bounded by y=e^x, the coordinate axes and the tangent at x=2.
ALSO
calculate the area bounded by y=e^x, the Y axis ansd the line y=e^2
plzzzz answer this as soon as possible
thanx alot

 

[Lazarus]

As with all questions of this nature, the first step you take should be to draw a graph.

Originally posted by skool
calculate the area bounded by y=e^x, the coordinate axes and the tangent at x=2.

Sketch y = e^x and derive the equation for the tangent when x = 2. Sketch that line too, and you'll see that it (rather nicely) intercepts the x-axis at x = 1.

You're after the area enclosed by y = e^x, the tangent and the axes. Simply find the area under y = e^x between x = 0 and x = 2, and subtract the area under the tangent between x = 1 and x = 2.

Originally posted by skool
calculate the area bounded by y=e^x, the Y axis ansd the line y=e^2

Add the line y = e² to your graph (where e² ~ 7.4). There are two methods you can employ for finding the area between y = e^x, the line and the y-axis.

1) Derive the inverse function and integrate it between y = 1 and y = e², or

2) Multiply e² by 2 to obtain the area of a rectangle, and then subtract the area under y = e^x between x = 0 and x = 2 (which you found in the first part of the question).

The latter is probably easier.

 

[skool :(]

one thing though

how would u integrate the area under the tangent?
the equation for the tangent is y=e^2(x-1).
plz can u actually do the workin out for the whole answer?
thanx

 

[Lazarus]

Re: one thing though

Originally posted by skool
how would u integrate the area under the tangent?
the equation for the tangent is y=e^2(x-1).
plz can u actually do the workin out for the whole answer?

It's just a straight line, and e² is just a constant. Integrate it in the normal way.<pre><font size=2>integral: e².(x - 1) dx
integral: e²x - e² dx
= [(e²/2)x² - e²x] from x = 1 to x = 2
= [(e²/2)(2)² - e²(2)] - [(e²/2)(1)² - e²(1)]
= 2e² - 2e² - e²/2 + e²
= e²/2 units²</font></pre>