Question

Really need help on this question!? Help!?

B) Approximate the area of each strip using circumscribed rectangle, and find the sum of the areas to those rectangles. How is that sum related to the exact area? Draw a diagram to support your answers.

C) Write the definite integral representing the exact area, and evaluate that integral.

Answer 1

By "inscribed rectangle" I think the paper is looking for rectangles that stay entirely "within" the function, that is, stay under the function entirely.

The best way to do this for each section would be with left-handed rectangles; that is, for example, in the `x=4` to `x=5` section drawing a rectangle with a height of `4` and width of `1`.

This can be done for each mini-section, with the exception of the very first section (the triangle, not a trapezoid) because a rectangle of any height would no longer be inscribed since the triangle begins at `(0,0)`. For this section you would just have a "rectangle" of area `0`, that is, a line.

The area of this section is `0(1)+1(1)+2(1)+3(1)+4(1)=10`.

As you can tell, this is an underestimate because there is part of the area of the region that the rectangles do not cover.

The exact area is given by `int_0^5xcolor(white).dx=[x^2/2]_0^5=25/2`.

Answer 2

B) 10, which is slightly less than the exact area.
C) `A=[1/2x^2]_0^5` = 12.5

Explanation:

Finding the area of a straight line using rectangles seems a bit odd, seeing as you can use the triangle to get an exact area `(½*base*height` = `½*5*5=12.5 units^2`)

But using the rectangles is the question so here we go.
I'm assuming inscribed area means underestimate, so turn each segment into a rectangles by drawing a line from the y=x graph to your rectangles. (sorry this is easy to draw but hard to explain and the image insert function doesn't seem to be working)

Then find the area of each rectangle using base times height and add them together:
`(1*0)+(1*1)+(1*2)+(1*3)+(1*4)`
`0+1+2+3+4`
=10 units squared.

Note that this is slightly less than the exact value I put above - this is because I've used inscribed rectangles, and there are little gaps above each one not included in my sum. You could get this closer to the actual value by making each rectangle slightly thinner.

Part C: definite integral
`Area=int_a^b("function")`

a=0, b= 5 (these are your boundaries)

`A=int_0^5f(x)dx`

`f(x) = x`

`A=[1/2x^2]_0^5`

`=(1/2(5)^2)-(1/2(0)^2)`

`=1/2(25)`

`=12.5` units squared

Hope this helps!