Question

Calculate the area of the shaded rectangles in Figure 18. Which approximation do these rectangles represent?

Answer 1

The area of a rectangle is just base times height. Just add all of their areas.

The type of approximation depends on where the top of each rectangle hits the curve. If the curve is in the middle of the top of the rectangle, it's midpoint. If it's the right or left corner, it's right or left riemann sum

Answer 2

you can find the height by plugging in the corresponding x value

Answer 3

so what do you get?

Answer 4

is it midpoint?

Answer 5

Yes it is midpoint.

Answer 6

so can you guide me through how to solve it using that method?

Answer 7

I'll walk you through the first rectangle. You need to find base times height. The base is easy, it's just the distance from -3 to -2, which is 1. The height is what you get when you plug the x value -1 into the equation. For that, you get

y(-1) = 4 + 1 / 1+1 = 5/2

so the area is 1 * 5/2 = 5/2. Do that for all rectangles and add all of the areas

Answer 8

dang did it wrong hold on the x value isnt -1

Answer 9

the x value you plug into the equation should be -2.5

Answer 10

dont ujust do the midpoint equation with F(n)+f(n+1)/2

Answer 11

confused

Answer 12

or do we not approximate?

Answer 13

can you just do it for me haha.

Answer 14

this method is a way to find the approximate area under the curve

Answer 15

so the answer will be 1 * (y(-2.5)+y(-1.5)+y(-.5)+y(.5)+y(1.5)+y(2.5)) just plug all those values into the equation

Answer 16

why is it 2.5?

Answer 17

For the first rectangle, the height will be y when you plug in halfway between -3 and -2, which is -2.5. If you plug -2.5 into the equation, you get the height of the rectangle.

Answer 18

oh I gotcha

Answer 19

What solution did you end up getting?

Answer 20

?

Answer 21

for the first rectangle or all of them?

Answer 22

all of them

Answer 23

?