Question

Use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). (Round your answers to three decimalplaces.)
y = 4e^−x

upper sum__________ lower sum___________

Thanks for the help!!! :)

Answer 1

4.3
2.1

Answer 2

y = sqrt (1-x^2)

Each sub interval needs to be 0.2 units

Upper sums
y(0) * 0.2 + y(0.2)*0.2 + y(0.4) *0.2 + y(0.6) * 0.2 + y(0.8) * 0.2
= (1/5)(y(0) + y(0.2) + y(0.4) + y(0.6) + y(0.8))

Lower sums
= (1/5)(y(0.2) + y(0.4) + y(0.6) + y(0.8) + y(1))

y(0) = 1
y(0.2) = sqrt (1-1/25) = sqrt (24/25)
y(0.4) = sqrt (1-4/25) = sqrt (21/25)
y(0.6) = sqrt (1-9/25) = sqrt (16/25) = 4/5
y(0.8) = sqrt (1-16/25) = sqrt (9/25) = 3/5
y(1) = 0
Upper sums =
0.8592622 units^2 (7dp)
Lower sums =
0.6592622 units^2 (7dp)

If you want to average these
A = 0.7592622 units^2 (7dp)


Since its 1/4 of a circle with radius = 1
Area = pi*162 / 4
=0.7853981634 units^2 (10dp)