Question

Use the rectangles in each graph to approximate the area of the region bounded by y=sin(x), y=0, x=0, x=pi

Answer 1

without the graph Im afraid we cannot help

Answer 2

im trying to get the graphs

Answer 3

how is that?

Answer 4

Number one. Lets do this

Answer 5

So you can see the scale. From 0 to 1 there are 5 equidistant markers. Thus 1/5=0.2 for each scale.

Answer 6

on the y axis?

Answer 7

That is on the y axis.
On the x axis, first image there is pi/4 scales.

Answer 8

ok

Answer 9

The first triangle goes up to 3.5 scale=3.5*.0.2=0.7 and it goes to the right by pi/4.
So the area is pi/4 * 0.7

Answer 10

What would be the second rectangles area? ( sorry I put triangle before by accident)

Answer 11

why did you multiply 3.5 by 0.2?

Answer 12

Because the first rectangles goes up to 3.5 scales. Thus one side is 3.5*0.2 long

Answer 13

oh ok

Answer 14

The other side has length pi/4. The area =0.7 *pi/4 (1st rectangle)

Answer 15

so the second rectangle is 1*pi/4=pi/4?

Answer 16

Correct

Answer 17

and the first is 0.7pi/4

Answer 18

And the third is equal to the first.
So if you add them up you will get the answer.

What is different in the second picture?

Answer 19

there are 5 rectangles

Answer 20

Yes, because the scale on the x axis is different.
What is it?

Answer 21

pi/6

Answer 22

Correct again.
I think now you have every tool to finish this question

Answer 23

also i think the area for the first graph is 3.4pi/4

Answer 24

im not sure if thats write

Answer 25

oops right*

Answer 26

0.7 *pi/4 +1*pi/4+0.7*pi/4=(0.7+1+.0.7)*pi/4=2.4*pi/4

Answer 27

oh i see where i went wrong

Answer 28

Probably it is neater to present this solution like this:
\[0.6\pi \]

Answer 29

thank you so much

Answer 30

For your interest the exact solution, so the area under this graph from 0 to pi is equal to 1.

Answer 31

Oh sorry that was half of it :)
It is equal to 2.

Answer 32

it is?

Answer 33

We got 0.6pi=1.885
So it is fairly good approximation. If you do the 2nd picture, the result should be closer to 2.

Answer 34

Yes it is, that area =2

Answer 35

yes it is :)

Answer 36

It is a better approximation. If you split the area into 1000000000000 rectangles than the answer might be 1.999999999

Answer 37

This is where Newton and Leibniz invented Calculus.
If you split the graph into infinite many rectangles than the area will be 2!

Answer 38

well you learn something new everyday

Answer 39

:)